Research ArticleOuter Synchronization of Simple Firefly DiscreteModels in Coupled Networks

A Arellano-Delgado1 C Cruz-Hernaacutendez1

R M Loacutepez Gutieacuterrez2 and C Posadas-Castillo3

1Electronics and Telecommunications Department Scientific Research and Advanced Studies Center of Ensenada (CICESE)22860 Ensenada BC Mexico2Engineering Architecture and Design Faculty Baja California Autonomous University (UABC) 22860 Ensenada BC Mexico3Mechanics and Electrical Engineering Faculty Nuevo Leon Autonomous University (UANL) 66450 San Nicolas de los GarzaNL Mexico

Correspondence should be addressed to C Cruz-Hernandez ccruzcicesemx

Received 31 December 2014 Accepted 1 May 2015

Academic Editor Xinkai Chen

Copyright copy 2015 A Arellano-Delgado et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Synchronization is one of the most important emerging collective behaviors in nature which results from the interaction ingroups of organisms In this paper network synchronization of discrete-time dynamical systems is studied In particular networksynchronization with fireflies oscillators like nodes is achieved by using complex systems theory Different cases of interest onnetwork synchronization are studied including for a large number of fireflies oscillators we consider synchronization in small-world networks and outer synchronization among different coupled networks topologies for all presented cases we provideappropriate ranges of values for coupling strength and extensive numerical simulations are included In addition for illustrativepurposes we show the effectiveness of network synchronization by means of experimental implementation of coupled nineelectronics fireflies in different topologies

1 Introduction

In nature several complex behaviors emerge from the inter-action among living organisms to generate a common ben-efit for example synchronization in shoaling fish to avoidpredators organization of ants to build the nest flock of birdsand pod of dolphins and one of the most spectacular syn-chronization cases is the firefly flashing in Southeast Asia andNorth America In [1 2] some studies on synchronization offireflies are reported Such insects use their bioluminescencefor mating purposes where thousands of male fireflies flashat the same time in a rhythmic form to attract the attentionof nearby females Synchronization was introduced by CHuygens when he was surprised with the synchronizationof two pendulum clocks coupled mutually by the wall anexperiment of Huygensrsquos clocks is studied in [3] In thelast decades synchronization has been studied on chemical

biological and ecological systems and electronics devices [4ndash7] Thus synchronization is the effect to produce the samedynamic behavior in two or more elements when they arecoupled with physical links in addition synchronizationcan be classified into two types mutual synchronization(bidirectional coupling) and master-slave synchronization(unidirectional coupling) these two types of synchronizationare widely discussed in [8 9]

The fireflies are considered as oscillators that are cou-pled by light pulses The coupling and synchronization ofoscillators have been studied by scientists since the 60s inbiological systems see for example [10ndash13] In 1990 Mirolloand Strogatz published an article where a mathematicalmodel is introduced to couple and synchronize an oscillatorby pulses in [14] they assume that all the oscillators in thesystem are identical this model was pioneer to explain thesynchronization in complex systems such as neurons cells

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 895379 14 pageshttpdxdoiorg1011552015895379

2 Mathematical Problems in Engineering

(0 1)

(12radic3

2)(radic2

2radic2

2)

(radic3

2 21)

(1 0)

(radic3

2minus1

2)

(radic2

2minusradic2

2)

(12minusradic3

2)

(0 minus1)(minus12minusradic3

2)

(minusradic2

2minusradic2

2)

(minusradic3

2minus1

2)

(minus1 0)

(minusradic3

21

2)

(minusradic2

2radic2

2)

(minus12radic3

2)

y

x

120587

2 120587

3 120587

4120587

6

11120587

67120587

4

7120587

6

5120587

3

2120587

3

5120587

4

5120587

6

4120587

3 3120587

2

3120587

4 90∘ 60∘

45∘

30∘

0∘

120∘

135∘

150∘

180∘

270∘

120587 0120579

300∘315∘

330∘210∘

225∘

240∘

Figure 1 The unitary circle that represents a circular unitary motion in the fireflies

t(k)

x

0 02 04 06 08 1 12 14 16 18 2

1050

minus05minus1

(a)

14

135

13

125

12

115

w

t(k)

0 02 04 06 08 1 12 14 16 18 2

(b)

1

05

0

minus05

minus1

x(k)

x(k minus 1)

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

(c)

Figure 2 Behavior of an isolated firefly (7) (a) temporal dynamics 119909(119896) blinking two times per second (b) angular frequency 119908(119896) and (c)phase-portrait 119909(119896) versus 119909(119896 minus 1)

Mathematical Problems in Engineering 3

of the heart and fireflies Another work about the analysisof oscillators coupled by pulses was presented by Ernst et alin [15] based on the Mirollo-Strogatz model but with cou-pling cancellation other approximations tomodeling firefliessynchronization have been proposed in [16 17] Thereforethe study and analysis on synchronization of fireflies canhelp to understand the emergence of collective behavior inother organisms and their applications such as robotics andcommunications

In this paper we use a dynamical model to synchronizein several topologies based on the coupling matrix First themodel of a firefly is presented and simulated then five firefliesare coupled in nearest-neighbor network star network andsmall-world network where the synchronization is achievedin all the reported cases We use the MatLab software tosimulate the synchronization process that is verified withthe graphics of temporal dynamics phase and error amongthe nodes Additionally we extend the analysis to outersynchronization of two and three coupled networks we provethe effectiveness of the proposed synchronization schemewith numerical results and experimental implementationThis paper is organized as follows In Section 2 we give abrief review on complex dynamical networks In Section 3the proposed firefly dynamical discrete simple model ispresented Network synchronization of different topologies isshown in Section 4 In Section 5 the outer synchronizationof two and three networks is presented In Section 6 experi-mental implementation of fireflies is presented Finally someconclusions are mentioned in Section 7

2 Preliminaries

In this section we give a brief review on synchronization ofcomplex dynamical networks

21 Synchronization of Complex Networks We consider acomplex network is composed of 119873 identical nodes linearlyand diffusively coupled through the first state of each nodeIn this network each node constitutes an 119873-dimensionaldiscrete-time map The state equations of this network aredescribed by

x119894 (119896 + 1) = 119891 (x119894 (119896)) + u119894 (119896) 119894 = 1 2 119873 (1)

where x119894(119896) = (119909

1198941(119896) 1199091198942(119896) 119909119894119873(119896))119879isin R119873 are the state

variables of the node 119894 and u119894(119896) = (119906

1198941(119896) 0 0) isin R119873 isthe input signal of the node 119894 and is defined by

u119894 (119896) = 119888

119873

sum

119895=1119886119894119895Γx119895 (119896) 119894 = 1 2 119873 (2)

where the constant 119888 gt 0 represents the coupling strength ofthe complex network and Γ isin R119873times119873 is constant 0-1 matrixlinking coupled state variablesMeanwhileA = (119886

119894119895) isin R119873times119873

is the couplingmatrix which represents the coupling topologyof the complex network If there is a connection betweennode

N1

N2N3

N4

N5

Figure 3 Network of five coupled fireflies in nearest-neighbortopology

119894 and node 119895 then 119886119894119895= 1 otherwise 119886

119894119895= 0 for 119894 = 119895 The

diagonal elements of coupling matrix A are defined as

119886119894119894= minus

119873

sum

119895=1119895 =119894119886119894119895= minus

119873

sum

119895=1119895 =119894119886119895119894 119894 = 1 2 119873 (3)

If the degree of node 119894 is 119889119894 then 119886

119894119894= minus119889119894 119894 = 1 2 119873

Now suppose that the complex network is connectedwithout isolated clusters Then A is a symmetric irreduciblematrix In this case it can be shown that zero is an eigenvalueof A with multiplicity 1 and all the other eigenvalues ofA arestrictly negative see [18 19]

In accordance with [19] the complex network (1)-(2) issaid to achieve (asymptotically) synchronization if

x1 (119896) = x2 (119896) = sdot sdot sdot = x119873 (119896) as 119896 997888rarr infin (4)

Diffusive coupling condition (3) guarantees that the synchro-nization state is a solution s(119896) isin R119873 of an isolated nodethat is

s (119896 + 1) = 119891 (s (119896)) (5)

where s(119905) can be an equilibrium point a periodic orbit or achaotic attractor Thus stability of the synchronization state

x1 (119896) = x2 (119896) = sdot sdot sdot = x119873 (119896) = s (119896) (6)

of complex network (1)-(2) is determined by the dynamicsof an isolated node that is function 119891 and solution s(119896)the coupling strength 119888 the inner linking matrix Γ and thecoupling matrix A

3 Dynamics of a Simple Firefly Discrete Model

In this section a simple firefly discrete-time mathematicalmodel is described in order to be used as fundamental nodesto construct different coupled networks

The simple dynamical model of a firefly is expressed bymeans of the following discrete-time system based on [20]

119909 (119896 + 1) = sin (119908 (119896) 119905 (119896))

119905 (119896 + 1) = 119905 (119896) + 120587

1000

(7)

4 Mathematical Problems in Engineering

1050

minus05minus1

xi

0 02 04 06 08 1 212 14 16 18

t(k)

x1x2x3

x4x5

(a)

403020100

wi

0 02 04 06 08 1 212 14 16 18

t(k)

w1

w2

w3

w4

w5

(b)

210

minus1minus2x

1minusxi+1

0 02 04 06 08 1 212 14 16 18

x1 minus x2x1 minus x3

x1 minus x4x1 minus x5

t(k)

(c)

Figure 4 Temporal dynamics of (a) 119909119894(119896) and (b) 119908

119894(119896) with 119894 = 1 2 5 and error (c) 1199091(119896) minus 119909119894+1(119896) with 119894 = 1 2 4 in the uncoupled

nearest-neighbor network

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x2

x1

(a)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x3

x1

(b)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x4

x1

(c)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x5

x1

(d)

Figure 5 Phase diagrams (a) 1199091 versus 1199092 (b) 1199091 versus 1199093 (c) 1199091 versus 1199094 and (d) 1199091 versus 1199095 in the uncoupled nearest-neighbor network

where 119909(119896) is the simple representation of a firefly in discrete-time 119908(119896) = 2120587119891 is the angular frequency in radians with119891 the blink frequency 119905(119896) is the sampling time and 119896 is thenumber of iterations

Isolated firefly (7) does not receive stimulus from anyother firefly in the network and it is flashing with a periodof time of 119905 seconds If a firefly blinks with a certain periodwe take on the sequence of flashes as a circular unitary

Mathematical Problems in Engineering 5

0 02 04 06 08 1 212 14 16 18

t(k)

xi

10

minus1

x1x2x3

x4x5

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

wi

20

40

0

w1

w2

w3

w4

w5

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

x1minusxi+1 1

0minus1

x1 minus x2x1 minus x3

x1 minus x4x1 minus x5

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

w1minuswi+1 20

0minus20minus40

w1 minus w2

w1 minus w3

w1 minus w4

w1 minus w5

(d)

Figure 6 Temporal dynamics of (a) 119909119894(119896) and (b) 119908

119894(119896) with 119894 = 1 2 5 and errors (c) 1199091(119896) minus 119909119894+1(119896) and (d) 1199081(119896) minus 119908119894+1(119896) with

119894 = 1 2 4 in the coupled nearest-neighbor network with 119888 = 001

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x2

x1

(a)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x3

x1

(b)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x4

x1

(c)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x5

x1

(d)

Figure 7 Phase diagrams (a) 1199091 versus 1199092 (b) 1199091 versus 1199093 (c) 1199091 versus 1199094 and (d) 1199091 versus 1199095 in the coupled nearest-neighbor network

motion see Figure 1 where the angular position 120579 is a pointin the unitary circle and a short flash or pulse of the fireflyoccurs when a rotation is completed

Figure 2 shows the behavior of isolated firefly (7) forinitial conditions 119909(0) = 0 119905(0) = 0 119891 = 2Hz and 119896 = 600iterations (note that for appreciation purposes we use theinterpolation option of the Matlab simulation software)

4 Network Synchronization of CoupledFireflies in Different Connection Topologies

In this section network synchronization of coupled fireflies(7) in different topologies and the corresponding numericalresults are reported In the following cases of studywe assumethat all the fireflies are connected without self-loops and

6 Mathematical Problems in Engineering

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 8 Synchronization diagram for the nearest-neighbor net-work

N1N2

N3

N4

N5

Figure 9 Graph for a star network corresponding to A119904

without multiple edges between two fireflies To be consistentwith notation in complex network (1)-(2) we have consideredthe following change of variable

x119894= [

1199091198941

1199091198942] = [

1199081198941

1199091198942] = [

119908119894

119909119894

] (8)

41 Nearest-Neighbor Coupled Network First we considera complex network according to (1)-(2) composed of 119873 =

5 coupled fireflies periodic oscillators (7) as fundamentalnodes in nearest-neighbor coupled topology see Figure 3The corresponding state equations to this dynamical networkare arranged as follows the first firefly1198731 is given by

1199081 (119896 + 1) = 1199081 (119896) + 11990611 (119896)

1199091 (119896 + 1) = sin (1199081 (119896) 119905 (119896))

11990611 (119896) = 119888 (minus21199081 (119896) +1199082 (119896) +1199085 (119896))

(9)

the second firefly1198732 is given by

1199082 (119896 + 1) = 1199082 (119896) + 11990621 (119896)

1199092 (119896 + 1) = sin (1199082 (119896) 119905 (119896))

11990621 (119896) = 119888 (minus21199082 (119896) +1199081 (119896) +1199083 (119896))

(10)

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 10 Synchronization diagram for the star network

N1

N2N3

N4

N5

Figure 11 Graph for a small-world network corresponding to Asw

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 12 Synchronization diagram for the small-world network

the third firefly1198733 is given by

1199083 (119896 + 1) = 1199083 (119896) + 11990631 (119896)

1199093 (119896 + 1) = sin (1199083 (119896) 119905 (119896))

11990631 (119896) = 119888 (minus21199083 (119896) +1199082 (119896) +1199084 (119896))

(11)

Mathematical Problems in Engineering 7

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

xisw

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

40

0wisw

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

x1s

wminusx(i+

1)sw

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

0minus40

w1s

wminusw(i+

1)sw

(d)

Figure 13 Small-world network in mass

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

Figure 14 Graph of outer synchronization between two coupledcomplex networks with different topologies

the fourth firefly1198734 is given by

1199084 (119896 + 1) = 1199084 (119896) + 11990641 (119896)

1199094 (119896 + 1) = sin (1199084 (119896) 119905 (119896))

11990641 (119896) = 119888 (minus21199084 (119896) +1199083 (119896) +1199085 (119896))

(12)

and the fifth firefly1198735 is given by

1199085 (119896 + 1) = 1199085 (119896) + 11990651 (119896)

1199095 (119896 + 1) = sin (1199085 (119896) 119905 (119896))

11990651 (119896) = 119888 (minus21199085 (119896) +1199084 (119896) +1199081 (119896))

(13)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (14)

The corresponding coupling matrixA for the network shownin Figure 3 is given by

A =

[

[

[

[

[

[

[

[

[

minus2 1 0 0 11 minus2 1 0 00 1 minus2 1 00 0 1 minus2 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(15)

For illustrative purposes we present the scenario where theoscillators fireflies of the network are uncoupled (isolated

fireflies) From (9) to (28) by choosing 119888 = 0 the controlinputs 119906

1198941 = 0 therefore the five isolated fireflies of thenetwork are unsynchronized assuming different initial con-ditions Figures 4 and 5 show the collective behavior of fiveisolated fireflies oscillators temporal dynamics of states 119909

119894(119896)

and 119908119894(119896) for 119894 = 1 2 5 synchronization errors 1199091(119896) minus

119909119894+1(119896) for 119894 = 1 2 3 4 and phase diagrams among the

oscillators fireflies 1199091 versus 119909119894 for 119894 = 2 3 4 5 with initialconditions 1199081(0) = 21205872 1199082(0) = 21205875 1199083(0) = 212058711199084(0) = 21205876 1199085(0) = 21205874 1199091(0) = 1199092(0) = 1199093(0) =1199094(0) = 1199095(0) = 0 119905(0) = 0 and 119896 = 600 Considernow the scenario where the oscillators fireflies of the networkare coupled in nearest-neighbor topology (see Figure 3) Fornetwork synchronization purposes we increase the couplingstrength values of 119888 for 119888 gt 0 Let 119888 = 001 with the sameinitial conditions as well as in uncoupled nearest-neighbornetwork Figure 6 shows the temporal dynamics and errorswhere1199081(119896)minus119908119894+1(119896) is computed to observe the behavior ofthe error among the angular frequencies of the synchronizedoscillators

In Figure 7 we can appreciate the phase diagrams whereafter a transient the involved signals overlap in a line of45 degrees denoting synchronization From Figures 6 and 7we can deduce that with this coupling strength and initialconditions the network is synchronizedWe obtained a rangeof coupling strength 119888 numerically We compute (for 119896 =

5000 so the calculation range of 119888 is approximate) a sweepof 119888 from 0 to 1 at intervals of 00001 After removing thefirst 4000 iterations of each of the states we compare the errorbetween states 119909

119894+1(119896) and 1199091(119896) for 119894 = 1 2 3 4 where if thesum of absolute value of the errors is greater than 001 thenwe established that the network is not synchronized (ie after4000 iterations if (|(1199091(119896) minus 1199092(119896))| + |(1199091(119896) minus 1199093(119896))| +|(1199091(119896) minus 1199094(119896))| + |(1199091(119896) minus 1199095(119896))|) gt 001 we establishedthat there is no synchronization in the network) Figure 8shows a diagram that we called synchronization diagramwhere the value of 0 denotes no synchronization and thevalue of 1 denotes synchronization thus we can see the range00019 lt 119888 lt 05521 where the nodes in the network aresynchronized

8 Mathematical Problems in Engineering

10

minus1xixis

0 05 1 15 2 25 3 35

t(k)

xixis

(a)

6040200w

iwis

0 05 1 15 2 25 3 35

t(k)

wi

wis

(b)

20

minus2ximinusxis

0 05 1 15 2 25 3 35

t(k)

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

100

minus10wiminuswis

0 05 1 15 2 25 3 35

t(k)

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 15 Dynamics for two uncoupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and

errors (c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xi xis

(a)

0 05 1 15 2 25 3 35

t(k)

6040200w

iwis

wi wis

(b)

0 05 1 15 2 25 3 35

t(k)

0051

minus05ximinusxis

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

0 05 1 15 2 25 3 35

t(k)

10

0minus10w

iminuswis

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 16 Dynamics for two coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and errors

(c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

42 Star Coupled Network Now we consider a complexnetwork according to (1)-(2) composed of 119873 = 5 coupledfireflies periodic oscillators (7) as fundamental nodes instar coupled topology see Figure 9 The corresponding stateequations to this dynamical network are arranged as followsthe first firefly1198731 is given by

1199081119904 (119896 + 1) = 1199081119904 (119896) + 11990611119904 (119896)

1199091119904 (119896 + 1) = sin (1199081119904 (119896) 119905 (119896))

11990611119904 (119896) = 119888 (minus41199081119904 (119896) +1199082119904 (119896) + sdot sdot sdot +1199083119904 (119896)

+1199084119904 (119896) +1199085119904 (119896))

(16)

the second firefly1198732 is given by

1199082119904 (119896 + 1) = 1199082119904 (119896) + 11990621119904 (119896)

1199092119904 (119896 + 1) = sin (1199082119904 (119896) 119905 (119896))

11990621119904 (119896) = 119888 (minus1199082119904 (119896) +1199081119904 (119896))

(17)

the third firefly1198733 is given by

1199083119904 (119896 + 1) = 1199083119904 (119896) + 11990631119904 (119896)

1199093119904 (119896 + 1) = sin (1199083119904 (119896) 119905 (119896))

11990631119904 (119896) = 119888 (minus1199083119904 (119896) +1199081119904 (119896))

(18)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 17 Synchronization diagram for outer synchronization ofnearest-neighbor-star networks

the fourth firefly1198734 is given by

1199084119904 (119896 + 1) = 1199084119904 (119896) + 11990641119904 (119896)

1199094119904 (119896 + 1) = sin (1199084119904 (119896) 119905 (119896))

11990641119904 (119896) = 119888 (minus1199084119904 (119896) +1199081119904 (119896))

(19)

and the fifth firefly1198735 is given by

1199085119904 (119896 + 1) = 1199085119904 (119896) + 11990651119904 (119896)

1199095119904 (119896 + 1) = sin (1199085119904 (119896) 119905 (119896))

11990651119904 (119896) = 119888 (minus1199085119904 (119896) +1199081119904 (119896))

(20)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (21)

The corresponding coupling matrix A119904 for the networkshown in Figure 9 is given by

A119904 =

[

[

[

[

[

[

[

[

[

minus4 1 1 1 11 minus1 0 0 01 0 minus1 0 01 0 0 minus1 01 0 0 0 minus1

]

]

]

]

]

]

]

]

]

(22)

where subscript 119904 refers to the star network We carry out thenumerical simulations in the star network with 119888 = 001 andinitial conditions 1199081119904(0) = 212058725 1199082119904(0) = 212058755 1199083119904(0) =212058715 1199084119904(0) = 212058765 1199085119904(0) = 212058745 1199091119904(0) = 1199092119904(0) =1199093119904(0) = 1199094119904(0) = 1199095119904(0) = 0 119905(0) = 0 and 119896 = 600 Takinginto account the above conditions we obtain similar results asin the nearest-neighbor network and synchronization amongthe nodes was achieved Figure 10 shows the synchronizationdiagram where we can see the range 00021 lt 119888 lt 03995where the nodes in the network are synchronized

43 Small-World Coupled Network A network with complexcoupled topology can be represented by a random graph butour intuition tells us clearly that many real complex networksare neither completely randomnor completely regular Some-thing that we must emphasize is that most of the biologicaltechnological and social networks lie between these twoextremesThese systems are highly clustered as regular arrayshowever they have a characteristic path length like randomnetworks see [21] These types of networks are called small-world networks in analogy to the small-world phenomenon(popularly known as six degrees of separation)

In general terms these networks are regular arrange-ments made by connections or reconnections of pairs ofrandomly chosen nodes Even a very small number ofthese added connections commonly called shortcuts do notchange the local properties (very high clustering typical of aregular network) and cause those typical random networksvalues to be present in the average path length see [22]

Now we consider a complex network according to (1)-(2) composed of 119873 = 5 coupled fireflies periodic oscillators(7) as fundamental nodes in small-world coupled topologysee Figure 11 The corresponding state equations to thisdynamical network are arranged as follows the first firefly1198731is given by1199081sw (119896 + 1) = 1199081sw (119896) + 11990611sw (119896)

1199091sw (119896 + 1) = sin (1199081sw (119896) 119905 (119896))

11990611sw (119896) = 119888 (minus31199081sw (119896) +1199082sw (119896) +1199084sw (119896) + sdot sdot sdot

+1199085sw (119896))

(23)

the second firefly1198732 is given by1199082sw (119896 + 1) = 1199082sw (119896) + 11990621sw (119896)

1199092sw (119896 + 1) = sin (1199082sw (119896) 119905 (119896))

11990621sw (119896) = 119888 (minus21199082sw (119896) +1199081sw (119896) +1199083sw (119896))

(24)

the third firefly1198733 is given by1199083sw (119896 + 1) = 1199083sw (119896) + 11990631sw (119896)

1199093sw (119896 + 1) = sin (1199083sw (119896) 119905 (119896))

11990631sw (119896) = 119888 (minus21199083sw (119896) +1199082sw (119896) +1199084sw (119896))

(25)

the fourth firefly1198734 is given by1199084sw (119896 + 1) = 1199084sw (119896) + 11990641sw (119896)

1199094sw (119896 + 1) = sin (1199084sw (119896) 119905 (119896))

11990641sw (119896) = 119888 (minus31199084sw (119896) +1199081sw (119896) +1199083sw (119896) + sdot sdot sdot

+1199085sw (119896))

(26)

and the fifth firefly1198735 is given by1199085sw (119896 + 1) = 1199085sw (119896) + 11990651sw (119896)

1199095sw (119896 + 1) = sin (1199085sw (119896) 119905 (119896))

11990651sw (119896) = 119888 (minus21199085sw (119896) +1199084sw (119896) +1199081sw (119896))

(27)

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

(0 1)

(12radic3

2)(radic2

2radic2

2)

(radic3

2 21)

(1 0)

(radic3

2minus1

2)

(radic2

2minusradic2

2)

(12minusradic3

2)

(0 minus1)(minus12minusradic3

2)

(minusradic2

2minusradic2

2)

(minusradic3

2minus1

2)

(minus1 0)

(minusradic3

21

2)

(minusradic2

2radic2

2)

(minus12radic3

2)

y

x

120587

2 120587

3 120587

4120587

6

11120587

67120587

4

7120587

6

5120587

3

2120587

3

5120587

4

5120587

6

4120587

3 3120587

2

3120587

4 90∘ 60∘

45∘

30∘

0∘

120∘

135∘

150∘

180∘

270∘

120587 0120579

300∘315∘

330∘210∘

225∘

240∘

Figure 1 The unitary circle that represents a circular unitary motion in the fireflies

t(k)

x

0 02 04 06 08 1 12 14 16 18 2

1050

minus05minus1

(a)

14

135

13

125

12

115

w

t(k)

0 02 04 06 08 1 12 14 16 18 2

(b)

1

05

0

minus05

minus1

x(k)

x(k minus 1)

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

(c)

Figure 2 Behavior of an isolated firefly (7) (a) temporal dynamics 119909(119896) blinking two times per second (b) angular frequency 119908(119896) and (c)phase-portrait 119909(119896) versus 119909(119896 minus 1)

Mathematical Problems in Engineering 3

of the heart and fireflies Another work about the analysisof oscillators coupled by pulses was presented by Ernst et alin [15] based on the Mirollo-Strogatz model but with cou-pling cancellation other approximations tomodeling firefliessynchronization have been proposed in [16 17] Thereforethe study and analysis on synchronization of fireflies canhelp to understand the emergence of collective behavior inother organisms and their applications such as robotics andcommunications

In this paper we use a dynamical model to synchronizein several topologies based on the coupling matrix First themodel of a firefly is presented and simulated then five firefliesare coupled in nearest-neighbor network star network andsmall-world network where the synchronization is achievedin all the reported cases We use the MatLab software tosimulate the synchronization process that is verified withthe graphics of temporal dynamics phase and error amongthe nodes Additionally we extend the analysis to outersynchronization of two and three coupled networks we provethe effectiveness of the proposed synchronization schemewith numerical results and experimental implementationThis paper is organized as follows In Section 2 we give abrief review on complex dynamical networks In Section 3the proposed firefly dynamical discrete simple model ispresented Network synchronization of different topologies isshown in Section 4 In Section 5 the outer synchronizationof two and three networks is presented In Section 6 experi-mental implementation of fireflies is presented Finally someconclusions are mentioned in Section 7

2 Preliminaries

In this section we give a brief review on synchronization ofcomplex dynamical networks

21 Synchronization of Complex Networks We consider acomplex network is composed of 119873 identical nodes linearlyand diffusively coupled through the first state of each nodeIn this network each node constitutes an 119873-dimensionaldiscrete-time map The state equations of this network aredescribed by

x119894 (119896 + 1) = 119891 (x119894 (119896)) + u119894 (119896) 119894 = 1 2 119873 (1)

where x119894(119896) = (119909

1198941(119896) 1199091198942(119896) 119909119894119873(119896))119879isin R119873 are the state

variables of the node 119894 and u119894(119896) = (119906

1198941(119896) 0 0) isin R119873 isthe input signal of the node 119894 and is defined by

u119894 (119896) = 119888

119873

sum

119895=1119886119894119895Γx119895 (119896) 119894 = 1 2 119873 (2)

where the constant 119888 gt 0 represents the coupling strength ofthe complex network and Γ isin R119873times119873 is constant 0-1 matrixlinking coupled state variablesMeanwhileA = (119886

119894119895) isin R119873times119873

is the couplingmatrix which represents the coupling topologyof the complex network If there is a connection betweennode

N1

N2N3

N4

N5

Figure 3 Network of five coupled fireflies in nearest-neighbortopology

119894 and node 119895 then 119886119894119895= 1 otherwise 119886

119894119895= 0 for 119894 = 119895 The

diagonal elements of coupling matrix A are defined as

119886119894119894= minus

119873

sum

119895=1119895 =119894119886119894119895= minus

119873

sum

119895=1119895 =119894119886119895119894 119894 = 1 2 119873 (3)

If the degree of node 119894 is 119889119894 then 119886

119894119894= minus119889119894 119894 = 1 2 119873

Now suppose that the complex network is connectedwithout isolated clusters Then A is a symmetric irreduciblematrix In this case it can be shown that zero is an eigenvalueof A with multiplicity 1 and all the other eigenvalues ofA arestrictly negative see [18 19]

In accordance with [19] the complex network (1)-(2) issaid to achieve (asymptotically) synchronization if

x1 (119896) = x2 (119896) = sdot sdot sdot = x119873 (119896) as 119896 997888rarr infin (4)

Diffusive coupling condition (3) guarantees that the synchro-nization state is a solution s(119896) isin R119873 of an isolated nodethat is

s (119896 + 1) = 119891 (s (119896)) (5)

where s(119905) can be an equilibrium point a periodic orbit or achaotic attractor Thus stability of the synchronization state

x1 (119896) = x2 (119896) = sdot sdot sdot = x119873 (119896) = s (119896) (6)

of complex network (1)-(2) is determined by the dynamicsof an isolated node that is function 119891 and solution s(119896)the coupling strength 119888 the inner linking matrix Γ and thecoupling matrix A

3 Dynamics of a Simple Firefly Discrete Model

In this section a simple firefly discrete-time mathematicalmodel is described in order to be used as fundamental nodesto construct different coupled networks

The simple dynamical model of a firefly is expressed bymeans of the following discrete-time system based on [20]

119909 (119896 + 1) = sin (119908 (119896) 119905 (119896))

119905 (119896 + 1) = 119905 (119896) + 120587

1000

(7)

4 Mathematical Problems in Engineering

1050

minus05minus1

xi

0 02 04 06 08 1 212 14 16 18

t(k)

x1x2x3

x4x5

(a)

403020100

wi

0 02 04 06 08 1 212 14 16 18

t(k)

w1

w2

w3

w4

w5

(b)

210

minus1minus2x

1minusxi+1

0 02 04 06 08 1 212 14 16 18

x1 minus x2x1 minus x3

x1 minus x4x1 minus x5

t(k)

(c)

Figure 4 Temporal dynamics of (a) 119909119894(119896) and (b) 119908

119894(119896) with 119894 = 1 2 5 and error (c) 1199091(119896) minus 119909119894+1(119896) with 119894 = 1 2 4 in the uncoupled

nearest-neighbor network

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x2

x1

(a)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x3

x1

(b)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x4

x1

(c)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x5

x1

(d)

Figure 5 Phase diagrams (a) 1199091 versus 1199092 (b) 1199091 versus 1199093 (c) 1199091 versus 1199094 and (d) 1199091 versus 1199095 in the uncoupled nearest-neighbor network

where 119909(119896) is the simple representation of a firefly in discrete-time 119908(119896) = 2120587119891 is the angular frequency in radians with119891 the blink frequency 119905(119896) is the sampling time and 119896 is thenumber of iterations

Isolated firefly (7) does not receive stimulus from anyother firefly in the network and it is flashing with a periodof time of 119905 seconds If a firefly blinks with a certain periodwe take on the sequence of flashes as a circular unitary

Mathematical Problems in Engineering 5

0 02 04 06 08 1 212 14 16 18

t(k)

xi

10

minus1

x1x2x3

x4x5

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

wi

20

40

0

w1

w2

w3

w4

w5

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

x1minusxi+1 1

0minus1

x1 minus x2x1 minus x3

x1 minus x4x1 minus x5

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

w1minuswi+1 20

0minus20minus40

w1 minus w2

w1 minus w3

w1 minus w4

w1 minus w5

(d)

Figure 6 Temporal dynamics of (a) 119909119894(119896) and (b) 119908

119894(119896) with 119894 = 1 2 5 and errors (c) 1199091(119896) minus 119909119894+1(119896) and (d) 1199081(119896) minus 119908119894+1(119896) with

119894 = 1 2 4 in the coupled nearest-neighbor network with 119888 = 001

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x2

x1

(a)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x3

x1

(b)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x4

x1

(c)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x5

x1

(d)

Figure 7 Phase diagrams (a) 1199091 versus 1199092 (b) 1199091 versus 1199093 (c) 1199091 versus 1199094 and (d) 1199091 versus 1199095 in the coupled nearest-neighbor network

motion see Figure 1 where the angular position 120579 is a pointin the unitary circle and a short flash or pulse of the fireflyoccurs when a rotation is completed

Figure 2 shows the behavior of isolated firefly (7) forinitial conditions 119909(0) = 0 119905(0) = 0 119891 = 2Hz and 119896 = 600iterations (note that for appreciation purposes we use theinterpolation option of the Matlab simulation software)

4 Network Synchronization of CoupledFireflies in Different Connection Topologies

In this section network synchronization of coupled fireflies(7) in different topologies and the corresponding numericalresults are reported In the following cases of studywe assumethat all the fireflies are connected without self-loops and

6 Mathematical Problems in Engineering

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 8 Synchronization diagram for the nearest-neighbor net-work

N1N2

N3

N4

N5

Figure 9 Graph for a star network corresponding to A119904

without multiple edges between two fireflies To be consistentwith notation in complex network (1)-(2) we have consideredthe following change of variable

x119894= [

1199091198941

1199091198942] = [

1199081198941

1199091198942] = [

119908119894

119909119894

] (8)

41 Nearest-Neighbor Coupled Network First we considera complex network according to (1)-(2) composed of 119873 =

5 coupled fireflies periodic oscillators (7) as fundamentalnodes in nearest-neighbor coupled topology see Figure 3The corresponding state equations to this dynamical networkare arranged as follows the first firefly1198731 is given by

1199081 (119896 + 1) = 1199081 (119896) + 11990611 (119896)

1199091 (119896 + 1) = sin (1199081 (119896) 119905 (119896))

11990611 (119896) = 119888 (minus21199081 (119896) +1199082 (119896) +1199085 (119896))

(9)

the second firefly1198732 is given by

1199082 (119896 + 1) = 1199082 (119896) + 11990621 (119896)

1199092 (119896 + 1) = sin (1199082 (119896) 119905 (119896))

11990621 (119896) = 119888 (minus21199082 (119896) +1199081 (119896) +1199083 (119896))

(10)

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 10 Synchronization diagram for the star network

N1

N2N3

N4

N5

Figure 11 Graph for a small-world network corresponding to Asw

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 12 Synchronization diagram for the small-world network

the third firefly1198733 is given by

1199083 (119896 + 1) = 1199083 (119896) + 11990631 (119896)

1199093 (119896 + 1) = sin (1199083 (119896) 119905 (119896))

11990631 (119896) = 119888 (minus21199083 (119896) +1199082 (119896) +1199084 (119896))

(11)

Mathematical Problems in Engineering 7

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

xisw

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

40

0wisw

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

x1s

wminusx(i+

1)sw

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

0minus40

w1s

wminusw(i+

1)sw

(d)

Figure 13 Small-world network in mass

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

Figure 14 Graph of outer synchronization between two coupledcomplex networks with different topologies

the fourth firefly1198734 is given by

1199084 (119896 + 1) = 1199084 (119896) + 11990641 (119896)

1199094 (119896 + 1) = sin (1199084 (119896) 119905 (119896))

11990641 (119896) = 119888 (minus21199084 (119896) +1199083 (119896) +1199085 (119896))

(12)

and the fifth firefly1198735 is given by

1199085 (119896 + 1) = 1199085 (119896) + 11990651 (119896)

1199095 (119896 + 1) = sin (1199085 (119896) 119905 (119896))

11990651 (119896) = 119888 (minus21199085 (119896) +1199084 (119896) +1199081 (119896))

(13)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (14)

The corresponding coupling matrixA for the network shownin Figure 3 is given by

A =

[

[

[

[

[

[

[

[

[

minus2 1 0 0 11 minus2 1 0 00 1 minus2 1 00 0 1 minus2 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(15)

For illustrative purposes we present the scenario where theoscillators fireflies of the network are uncoupled (isolated

fireflies) From (9) to (28) by choosing 119888 = 0 the controlinputs 119906

1198941 = 0 therefore the five isolated fireflies of thenetwork are unsynchronized assuming different initial con-ditions Figures 4 and 5 show the collective behavior of fiveisolated fireflies oscillators temporal dynamics of states 119909

119894(119896)

and 119908119894(119896) for 119894 = 1 2 5 synchronization errors 1199091(119896) minus

119909119894+1(119896) for 119894 = 1 2 3 4 and phase diagrams among the

oscillators fireflies 1199091 versus 119909119894 for 119894 = 2 3 4 5 with initialconditions 1199081(0) = 21205872 1199082(0) = 21205875 1199083(0) = 212058711199084(0) = 21205876 1199085(0) = 21205874 1199091(0) = 1199092(0) = 1199093(0) =1199094(0) = 1199095(0) = 0 119905(0) = 0 and 119896 = 600 Considernow the scenario where the oscillators fireflies of the networkare coupled in nearest-neighbor topology (see Figure 3) Fornetwork synchronization purposes we increase the couplingstrength values of 119888 for 119888 gt 0 Let 119888 = 001 with the sameinitial conditions as well as in uncoupled nearest-neighbornetwork Figure 6 shows the temporal dynamics and errorswhere1199081(119896)minus119908119894+1(119896) is computed to observe the behavior ofthe error among the angular frequencies of the synchronizedoscillators

In Figure 7 we can appreciate the phase diagrams whereafter a transient the involved signals overlap in a line of45 degrees denoting synchronization From Figures 6 and 7we can deduce that with this coupling strength and initialconditions the network is synchronizedWe obtained a rangeof coupling strength 119888 numerically We compute (for 119896 =

5000 so the calculation range of 119888 is approximate) a sweepof 119888 from 0 to 1 at intervals of 00001 After removing thefirst 4000 iterations of each of the states we compare the errorbetween states 119909

119894+1(119896) and 1199091(119896) for 119894 = 1 2 3 4 where if thesum of absolute value of the errors is greater than 001 thenwe established that the network is not synchronized (ie after4000 iterations if (|(1199091(119896) minus 1199092(119896))| + |(1199091(119896) minus 1199093(119896))| +|(1199091(119896) minus 1199094(119896))| + |(1199091(119896) minus 1199095(119896))|) gt 001 we establishedthat there is no synchronization in the network) Figure 8shows a diagram that we called synchronization diagramwhere the value of 0 denotes no synchronization and thevalue of 1 denotes synchronization thus we can see the range00019 lt 119888 lt 05521 where the nodes in the network aresynchronized

8 Mathematical Problems in Engineering

10

minus1xixis

0 05 1 15 2 25 3 35

t(k)

xixis

(a)

6040200w

iwis

0 05 1 15 2 25 3 35

t(k)

wi

wis

(b)

20

minus2ximinusxis

0 05 1 15 2 25 3 35

t(k)

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

100

minus10wiminuswis

0 05 1 15 2 25 3 35

t(k)

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 15 Dynamics for two uncoupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and

errors (c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xi xis

(a)

0 05 1 15 2 25 3 35

t(k)

6040200w

iwis

wi wis

(b)

0 05 1 15 2 25 3 35

t(k)

0051

minus05ximinusxis

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

0 05 1 15 2 25 3 35

t(k)

10

0minus10w

iminuswis

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 16 Dynamics for two coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and errors

(c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

42 Star Coupled Network Now we consider a complexnetwork according to (1)-(2) composed of 119873 = 5 coupledfireflies periodic oscillators (7) as fundamental nodes instar coupled topology see Figure 9 The corresponding stateequations to this dynamical network are arranged as followsthe first firefly1198731 is given by

1199081119904 (119896 + 1) = 1199081119904 (119896) + 11990611119904 (119896)

1199091119904 (119896 + 1) = sin (1199081119904 (119896) 119905 (119896))

11990611119904 (119896) = 119888 (minus41199081119904 (119896) +1199082119904 (119896) + sdot sdot sdot +1199083119904 (119896)

+1199084119904 (119896) +1199085119904 (119896))

(16)

the second firefly1198732 is given by

1199082119904 (119896 + 1) = 1199082119904 (119896) + 11990621119904 (119896)

1199092119904 (119896 + 1) = sin (1199082119904 (119896) 119905 (119896))

11990621119904 (119896) = 119888 (minus1199082119904 (119896) +1199081119904 (119896))

(17)

the third firefly1198733 is given by

1199083119904 (119896 + 1) = 1199083119904 (119896) + 11990631119904 (119896)

1199093119904 (119896 + 1) = sin (1199083119904 (119896) 119905 (119896))

11990631119904 (119896) = 119888 (minus1199083119904 (119896) +1199081119904 (119896))

(18)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 17 Synchronization diagram for outer synchronization ofnearest-neighbor-star networks

the fourth firefly1198734 is given by

1199084119904 (119896 + 1) = 1199084119904 (119896) + 11990641119904 (119896)

1199094119904 (119896 + 1) = sin (1199084119904 (119896) 119905 (119896))

11990641119904 (119896) = 119888 (minus1199084119904 (119896) +1199081119904 (119896))

(19)

and the fifth firefly1198735 is given by

1199085119904 (119896 + 1) = 1199085119904 (119896) + 11990651119904 (119896)

1199095119904 (119896 + 1) = sin (1199085119904 (119896) 119905 (119896))

11990651119904 (119896) = 119888 (minus1199085119904 (119896) +1199081119904 (119896))

(20)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (21)

The corresponding coupling matrix A119904 for the networkshown in Figure 9 is given by

A119904 =

[

[

[

[

[

[

[

[

[

minus4 1 1 1 11 minus1 0 0 01 0 minus1 0 01 0 0 minus1 01 0 0 0 minus1

]

]

]

]

]

]

]

]

]

(22)

where subscript 119904 refers to the star network We carry out thenumerical simulations in the star network with 119888 = 001 andinitial conditions 1199081119904(0) = 212058725 1199082119904(0) = 212058755 1199083119904(0) =212058715 1199084119904(0) = 212058765 1199085119904(0) = 212058745 1199091119904(0) = 1199092119904(0) =1199093119904(0) = 1199094119904(0) = 1199095119904(0) = 0 119905(0) = 0 and 119896 = 600 Takinginto account the above conditions we obtain similar results asin the nearest-neighbor network and synchronization amongthe nodes was achieved Figure 10 shows the synchronizationdiagram where we can see the range 00021 lt 119888 lt 03995where the nodes in the network are synchronized

43 Small-World Coupled Network A network with complexcoupled topology can be represented by a random graph butour intuition tells us clearly that many real complex networksare neither completely randomnor completely regular Some-thing that we must emphasize is that most of the biologicaltechnological and social networks lie between these twoextremesThese systems are highly clustered as regular arrayshowever they have a characteristic path length like randomnetworks see [21] These types of networks are called small-world networks in analogy to the small-world phenomenon(popularly known as six degrees of separation)

In general terms these networks are regular arrange-ments made by connections or reconnections of pairs ofrandomly chosen nodes Even a very small number ofthese added connections commonly called shortcuts do notchange the local properties (very high clustering typical of aregular network) and cause those typical random networksvalues to be present in the average path length see [22]

Now we consider a complex network according to (1)-(2) composed of 119873 = 5 coupled fireflies periodic oscillators(7) as fundamental nodes in small-world coupled topologysee Figure 11 The corresponding state equations to thisdynamical network are arranged as follows the first firefly1198731is given by1199081sw (119896 + 1) = 1199081sw (119896) + 11990611sw (119896)

1199091sw (119896 + 1) = sin (1199081sw (119896) 119905 (119896))

11990611sw (119896) = 119888 (minus31199081sw (119896) +1199082sw (119896) +1199084sw (119896) + sdot sdot sdot

+1199085sw (119896))

(23)

the second firefly1198732 is given by1199082sw (119896 + 1) = 1199082sw (119896) + 11990621sw (119896)

1199092sw (119896 + 1) = sin (1199082sw (119896) 119905 (119896))

11990621sw (119896) = 119888 (minus21199082sw (119896) +1199081sw (119896) +1199083sw (119896))

(24)

the third firefly1198733 is given by1199083sw (119896 + 1) = 1199083sw (119896) + 11990631sw (119896)

1199093sw (119896 + 1) = sin (1199083sw (119896) 119905 (119896))

11990631sw (119896) = 119888 (minus21199083sw (119896) +1199082sw (119896) +1199084sw (119896))

(25)

the fourth firefly1198734 is given by1199084sw (119896 + 1) = 1199084sw (119896) + 11990641sw (119896)

1199094sw (119896 + 1) = sin (1199084sw (119896) 119905 (119896))

11990641sw (119896) = 119888 (minus31199084sw (119896) +1199081sw (119896) +1199083sw (119896) + sdot sdot sdot

+1199085sw (119896))

(26)

and the fifth firefly1198735 is given by1199085sw (119896 + 1) = 1199085sw (119896) + 11990651sw (119896)

1199095sw (119896 + 1) = sin (1199085sw (119896) 119905 (119896))

11990651sw (119896) = 119888 (minus21199085sw (119896) +1199084sw (119896) +1199081sw (119896))

(27)

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

of the heart and fireflies Another work about the analysisof oscillators coupled by pulses was presented by Ernst et alin [15] based on the Mirollo-Strogatz model but with cou-pling cancellation other approximations tomodeling firefliessynchronization have been proposed in [16 17] Thereforethe study and analysis on synchronization of fireflies canhelp to understand the emergence of collective behavior inother organisms and their applications such as robotics andcommunications

In this paper we use a dynamical model to synchronizein several topologies based on the coupling matrix First themodel of a firefly is presented and simulated then five firefliesare coupled in nearest-neighbor network star network andsmall-world network where the synchronization is achievedin all the reported cases We use the MatLab software tosimulate the synchronization process that is verified withthe graphics of temporal dynamics phase and error amongthe nodes Additionally we extend the analysis to outersynchronization of two and three coupled networks we provethe effectiveness of the proposed synchronization schemewith numerical results and experimental implementationThis paper is organized as follows In Section 2 we give abrief review on complex dynamical networks In Section 3the proposed firefly dynamical discrete simple model ispresented Network synchronization of different topologies isshown in Section 4 In Section 5 the outer synchronizationof two and three networks is presented In Section 6 experi-mental implementation of fireflies is presented Finally someconclusions are mentioned in Section 7

2 Preliminaries

In this section we give a brief review on synchronization ofcomplex dynamical networks

21 Synchronization of Complex Networks We consider acomplex network is composed of 119873 identical nodes linearlyand diffusively coupled through the first state of each nodeIn this network each node constitutes an 119873-dimensionaldiscrete-time map The state equations of this network aredescribed by

x119894 (119896 + 1) = 119891 (x119894 (119896)) + u119894 (119896) 119894 = 1 2 119873 (1)

where x119894(119896) = (119909

1198941(119896) 1199091198942(119896) 119909119894119873(119896))119879isin R119873 are the state

variables of the node 119894 and u119894(119896) = (119906

1198941(119896) 0 0) isin R119873 isthe input signal of the node 119894 and is defined by

u119894 (119896) = 119888

119873

sum

119895=1119886119894119895Γx119895 (119896) 119894 = 1 2 119873 (2)

where the constant 119888 gt 0 represents the coupling strength ofthe complex network and Γ isin R119873times119873 is constant 0-1 matrixlinking coupled state variablesMeanwhileA = (119886

119894119895) isin R119873times119873

is the couplingmatrix which represents the coupling topologyof the complex network If there is a connection betweennode

N1

N2N3

N4

N5

Figure 3 Network of five coupled fireflies in nearest-neighbortopology

119894 and node 119895 then 119886119894119895= 1 otherwise 119886

119894119895= 0 for 119894 = 119895 The

diagonal elements of coupling matrix A are defined as

119886119894119894= minus

119873

sum

119895=1119895 =119894119886119894119895= minus

119873

sum

119895=1119895 =119894119886119895119894 119894 = 1 2 119873 (3)

If the degree of node 119894 is 119889119894 then 119886

119894119894= minus119889119894 119894 = 1 2 119873

Now suppose that the complex network is connectedwithout isolated clusters Then A is a symmetric irreduciblematrix In this case it can be shown that zero is an eigenvalueof A with multiplicity 1 and all the other eigenvalues ofA arestrictly negative see [18 19]

In accordance with [19] the complex network (1)-(2) issaid to achieve (asymptotically) synchronization if

x1 (119896) = x2 (119896) = sdot sdot sdot = x119873 (119896) as 119896 997888rarr infin (4)

Diffusive coupling condition (3) guarantees that the synchro-nization state is a solution s(119896) isin R119873 of an isolated nodethat is

s (119896 + 1) = 119891 (s (119896)) (5)

where s(119905) can be an equilibrium point a periodic orbit or achaotic attractor Thus stability of the synchronization state

x1 (119896) = x2 (119896) = sdot sdot sdot = x119873 (119896) = s (119896) (6)

of complex network (1)-(2) is determined by the dynamicsof an isolated node that is function 119891 and solution s(119896)the coupling strength 119888 the inner linking matrix Γ and thecoupling matrix A

3 Dynamics of a Simple Firefly Discrete Model

In this section a simple firefly discrete-time mathematicalmodel is described in order to be used as fundamental nodesto construct different coupled networks

The simple dynamical model of a firefly is expressed bymeans of the following discrete-time system based on [20]

119909 (119896 + 1) = sin (119908 (119896) 119905 (119896))

119905 (119896 + 1) = 119905 (119896) + 120587

1000

(7)

4 Mathematical Problems in Engineering

1050

minus05minus1

xi

0 02 04 06 08 1 212 14 16 18

t(k)

x1x2x3

x4x5

(a)

403020100

wi

0 02 04 06 08 1 212 14 16 18

t(k)

w1

w2

w3

w4

w5

(b)

210

minus1minus2x

1minusxi+1

0 02 04 06 08 1 212 14 16 18

x1 minus x2x1 minus x3

x1 minus x4x1 minus x5

t(k)

(c)

Figure 4 Temporal dynamics of (a) 119909119894(119896) and (b) 119908

119894(119896) with 119894 = 1 2 5 and error (c) 1199091(119896) minus 119909119894+1(119896) with 119894 = 1 2 4 in the uncoupled

nearest-neighbor network

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x2

x1

(a)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x3

x1

(b)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x4

x1

(c)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x5

x1

(d)

Figure 5 Phase diagrams (a) 1199091 versus 1199092 (b) 1199091 versus 1199093 (c) 1199091 versus 1199094 and (d) 1199091 versus 1199095 in the uncoupled nearest-neighbor network

where 119909(119896) is the simple representation of a firefly in discrete-time 119908(119896) = 2120587119891 is the angular frequency in radians with119891 the blink frequency 119905(119896) is the sampling time and 119896 is thenumber of iterations

Isolated firefly (7) does not receive stimulus from anyother firefly in the network and it is flashing with a periodof time of 119905 seconds If a firefly blinks with a certain periodwe take on the sequence of flashes as a circular unitary

Mathematical Problems in Engineering 5

0 02 04 06 08 1 212 14 16 18

t(k)

xi

10

minus1

x1x2x3

x4x5

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

wi

20

40

0

w1

w2

w3

w4

w5

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

x1minusxi+1 1

0minus1

x1 minus x2x1 minus x3

x1 minus x4x1 minus x5

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

w1minuswi+1 20

0minus20minus40

w1 minus w2

w1 minus w3

w1 minus w4

w1 minus w5

(d)

Figure 6 Temporal dynamics of (a) 119909119894(119896) and (b) 119908

119894(119896) with 119894 = 1 2 5 and errors (c) 1199091(119896) minus 119909119894+1(119896) and (d) 1199081(119896) minus 119908119894+1(119896) with

119894 = 1 2 4 in the coupled nearest-neighbor network with 119888 = 001

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x2

x1

(a)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x3

x1

(b)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x4

x1

(c)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x5

x1

(d)

Figure 7 Phase diagrams (a) 1199091 versus 1199092 (b) 1199091 versus 1199093 (c) 1199091 versus 1199094 and (d) 1199091 versus 1199095 in the coupled nearest-neighbor network

motion see Figure 1 where the angular position 120579 is a pointin the unitary circle and a short flash or pulse of the fireflyoccurs when a rotation is completed

Figure 2 shows the behavior of isolated firefly (7) forinitial conditions 119909(0) = 0 119905(0) = 0 119891 = 2Hz and 119896 = 600iterations (note that for appreciation purposes we use theinterpolation option of the Matlab simulation software)

4 Network Synchronization of CoupledFireflies in Different Connection Topologies

In this section network synchronization of coupled fireflies(7) in different topologies and the corresponding numericalresults are reported In the following cases of studywe assumethat all the fireflies are connected without self-loops and

6 Mathematical Problems in Engineering

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 8 Synchronization diagram for the nearest-neighbor net-work

N1N2

N3

N4

N5

Figure 9 Graph for a star network corresponding to A119904

without multiple edges between two fireflies To be consistentwith notation in complex network (1)-(2) we have consideredthe following change of variable

x119894= [

1199091198941

1199091198942] = [

1199081198941

1199091198942] = [

119908119894

119909119894

] (8)

41 Nearest-Neighbor Coupled Network First we considera complex network according to (1)-(2) composed of 119873 =

5 coupled fireflies periodic oscillators (7) as fundamentalnodes in nearest-neighbor coupled topology see Figure 3The corresponding state equations to this dynamical networkare arranged as follows the first firefly1198731 is given by

1199081 (119896 + 1) = 1199081 (119896) + 11990611 (119896)

1199091 (119896 + 1) = sin (1199081 (119896) 119905 (119896))

11990611 (119896) = 119888 (minus21199081 (119896) +1199082 (119896) +1199085 (119896))

(9)

the second firefly1198732 is given by

1199082 (119896 + 1) = 1199082 (119896) + 11990621 (119896)

1199092 (119896 + 1) = sin (1199082 (119896) 119905 (119896))

11990621 (119896) = 119888 (minus21199082 (119896) +1199081 (119896) +1199083 (119896))

(10)

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 10 Synchronization diagram for the star network

N1

N2N3

N4

N5

Figure 11 Graph for a small-world network corresponding to Asw

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 12 Synchronization diagram for the small-world network

the third firefly1198733 is given by

1199083 (119896 + 1) = 1199083 (119896) + 11990631 (119896)

1199093 (119896 + 1) = sin (1199083 (119896) 119905 (119896))

11990631 (119896) = 119888 (minus21199083 (119896) +1199082 (119896) +1199084 (119896))

(11)

Mathematical Problems in Engineering 7

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

xisw

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

40

0wisw

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

x1s

wminusx(i+

1)sw

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

0minus40

w1s

wminusw(i+

1)sw

(d)

Figure 13 Small-world network in mass

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

Figure 14 Graph of outer synchronization between two coupledcomplex networks with different topologies

the fourth firefly1198734 is given by

1199084 (119896 + 1) = 1199084 (119896) + 11990641 (119896)

1199094 (119896 + 1) = sin (1199084 (119896) 119905 (119896))

11990641 (119896) = 119888 (minus21199084 (119896) +1199083 (119896) +1199085 (119896))

(12)

and the fifth firefly1198735 is given by

1199085 (119896 + 1) = 1199085 (119896) + 11990651 (119896)

1199095 (119896 + 1) = sin (1199085 (119896) 119905 (119896))

11990651 (119896) = 119888 (minus21199085 (119896) +1199084 (119896) +1199081 (119896))

(13)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (14)

The corresponding coupling matrixA for the network shownin Figure 3 is given by

A =

[

[

[

[

[

[

[

[

[

minus2 1 0 0 11 minus2 1 0 00 1 minus2 1 00 0 1 minus2 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(15)

For illustrative purposes we present the scenario where theoscillators fireflies of the network are uncoupled (isolated

fireflies) From (9) to (28) by choosing 119888 = 0 the controlinputs 119906

1198941 = 0 therefore the five isolated fireflies of thenetwork are unsynchronized assuming different initial con-ditions Figures 4 and 5 show the collective behavior of fiveisolated fireflies oscillators temporal dynamics of states 119909

119894(119896)

and 119908119894(119896) for 119894 = 1 2 5 synchronization errors 1199091(119896) minus

119909119894+1(119896) for 119894 = 1 2 3 4 and phase diagrams among the

oscillators fireflies 1199091 versus 119909119894 for 119894 = 2 3 4 5 with initialconditions 1199081(0) = 21205872 1199082(0) = 21205875 1199083(0) = 212058711199084(0) = 21205876 1199085(0) = 21205874 1199091(0) = 1199092(0) = 1199093(0) =1199094(0) = 1199095(0) = 0 119905(0) = 0 and 119896 = 600 Considernow the scenario where the oscillators fireflies of the networkare coupled in nearest-neighbor topology (see Figure 3) Fornetwork synchronization purposes we increase the couplingstrength values of 119888 for 119888 gt 0 Let 119888 = 001 with the sameinitial conditions as well as in uncoupled nearest-neighbornetwork Figure 6 shows the temporal dynamics and errorswhere1199081(119896)minus119908119894+1(119896) is computed to observe the behavior ofthe error among the angular frequencies of the synchronizedoscillators

In Figure 7 we can appreciate the phase diagrams whereafter a transient the involved signals overlap in a line of45 degrees denoting synchronization From Figures 6 and 7we can deduce that with this coupling strength and initialconditions the network is synchronizedWe obtained a rangeof coupling strength 119888 numerically We compute (for 119896 =

5000 so the calculation range of 119888 is approximate) a sweepof 119888 from 0 to 1 at intervals of 00001 After removing thefirst 4000 iterations of each of the states we compare the errorbetween states 119909

119894+1(119896) and 1199091(119896) for 119894 = 1 2 3 4 where if thesum of absolute value of the errors is greater than 001 thenwe established that the network is not synchronized (ie after4000 iterations if (|(1199091(119896) minus 1199092(119896))| + |(1199091(119896) minus 1199093(119896))| +|(1199091(119896) minus 1199094(119896))| + |(1199091(119896) minus 1199095(119896))|) gt 001 we establishedthat there is no synchronization in the network) Figure 8shows a diagram that we called synchronization diagramwhere the value of 0 denotes no synchronization and thevalue of 1 denotes synchronization thus we can see the range00019 lt 119888 lt 05521 where the nodes in the network aresynchronized

8 Mathematical Problems in Engineering

10

minus1xixis

0 05 1 15 2 25 3 35

t(k)

xixis

(a)

6040200w

iwis

0 05 1 15 2 25 3 35

t(k)

wi

wis

(b)

20

minus2ximinusxis

0 05 1 15 2 25 3 35

t(k)

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

100

minus10wiminuswis

0 05 1 15 2 25 3 35

t(k)

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 15 Dynamics for two uncoupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and

errors (c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xi xis

(a)

0 05 1 15 2 25 3 35

t(k)

6040200w

iwis

wi wis

(b)

0 05 1 15 2 25 3 35

t(k)

0051

minus05ximinusxis

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

0 05 1 15 2 25 3 35

t(k)

10

0minus10w

iminuswis

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 16 Dynamics for two coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and errors

(c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

42 Star Coupled Network Now we consider a complexnetwork according to (1)-(2) composed of 119873 = 5 coupledfireflies periodic oscillators (7) as fundamental nodes instar coupled topology see Figure 9 The corresponding stateequations to this dynamical network are arranged as followsthe first firefly1198731 is given by

1199081119904 (119896 + 1) = 1199081119904 (119896) + 11990611119904 (119896)

1199091119904 (119896 + 1) = sin (1199081119904 (119896) 119905 (119896))

11990611119904 (119896) = 119888 (minus41199081119904 (119896) +1199082119904 (119896) + sdot sdot sdot +1199083119904 (119896)

+1199084119904 (119896) +1199085119904 (119896))

(16)

the second firefly1198732 is given by

1199082119904 (119896 + 1) = 1199082119904 (119896) + 11990621119904 (119896)

1199092119904 (119896 + 1) = sin (1199082119904 (119896) 119905 (119896))

11990621119904 (119896) = 119888 (minus1199082119904 (119896) +1199081119904 (119896))

(17)

the third firefly1198733 is given by

1199083119904 (119896 + 1) = 1199083119904 (119896) + 11990631119904 (119896)

1199093119904 (119896 + 1) = sin (1199083119904 (119896) 119905 (119896))

11990631119904 (119896) = 119888 (minus1199083119904 (119896) +1199081119904 (119896))

(18)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 17 Synchronization diagram for outer synchronization ofnearest-neighbor-star networks

the fourth firefly1198734 is given by

1199084119904 (119896 + 1) = 1199084119904 (119896) + 11990641119904 (119896)

1199094119904 (119896 + 1) = sin (1199084119904 (119896) 119905 (119896))

11990641119904 (119896) = 119888 (minus1199084119904 (119896) +1199081119904 (119896))

(19)

and the fifth firefly1198735 is given by

1199085119904 (119896 + 1) = 1199085119904 (119896) + 11990651119904 (119896)

1199095119904 (119896 + 1) = sin (1199085119904 (119896) 119905 (119896))

11990651119904 (119896) = 119888 (minus1199085119904 (119896) +1199081119904 (119896))

(20)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (21)

The corresponding coupling matrix A119904 for the networkshown in Figure 9 is given by

A119904 =

[

[

[

[

[

[

[

[

[

minus4 1 1 1 11 minus1 0 0 01 0 minus1 0 01 0 0 minus1 01 0 0 0 minus1

]

]

]

]

]

]

]

]

]

(22)

where subscript 119904 refers to the star network We carry out thenumerical simulations in the star network with 119888 = 001 andinitial conditions 1199081119904(0) = 212058725 1199082119904(0) = 212058755 1199083119904(0) =212058715 1199084119904(0) = 212058765 1199085119904(0) = 212058745 1199091119904(0) = 1199092119904(0) =1199093119904(0) = 1199094119904(0) = 1199095119904(0) = 0 119905(0) = 0 and 119896 = 600 Takinginto account the above conditions we obtain similar results asin the nearest-neighbor network and synchronization amongthe nodes was achieved Figure 10 shows the synchronizationdiagram where we can see the range 00021 lt 119888 lt 03995where the nodes in the network are synchronized

43 Small-World Coupled Network A network with complexcoupled topology can be represented by a random graph butour intuition tells us clearly that many real complex networksare neither completely randomnor completely regular Some-thing that we must emphasize is that most of the biologicaltechnological and social networks lie between these twoextremesThese systems are highly clustered as regular arrayshowever they have a characteristic path length like randomnetworks see [21] These types of networks are called small-world networks in analogy to the small-world phenomenon(popularly known as six degrees of separation)

In general terms these networks are regular arrange-ments made by connections or reconnections of pairs ofrandomly chosen nodes Even a very small number ofthese added connections commonly called shortcuts do notchange the local properties (very high clustering typical of aregular network) and cause those typical random networksvalues to be present in the average path length see [22]

Now we consider a complex network according to (1)-(2) composed of 119873 = 5 coupled fireflies periodic oscillators(7) as fundamental nodes in small-world coupled topologysee Figure 11 The corresponding state equations to thisdynamical network are arranged as follows the first firefly1198731is given by1199081sw (119896 + 1) = 1199081sw (119896) + 11990611sw (119896)

1199091sw (119896 + 1) = sin (1199081sw (119896) 119905 (119896))

11990611sw (119896) = 119888 (minus31199081sw (119896) +1199082sw (119896) +1199084sw (119896) + sdot sdot sdot

+1199085sw (119896))

(23)

the second firefly1198732 is given by1199082sw (119896 + 1) = 1199082sw (119896) + 11990621sw (119896)

1199092sw (119896 + 1) = sin (1199082sw (119896) 119905 (119896))

11990621sw (119896) = 119888 (minus21199082sw (119896) +1199081sw (119896) +1199083sw (119896))

(24)

the third firefly1198733 is given by1199083sw (119896 + 1) = 1199083sw (119896) + 11990631sw (119896)

1199093sw (119896 + 1) = sin (1199083sw (119896) 119905 (119896))

11990631sw (119896) = 119888 (minus21199083sw (119896) +1199082sw (119896) +1199084sw (119896))

(25)

the fourth firefly1198734 is given by1199084sw (119896 + 1) = 1199084sw (119896) + 11990641sw (119896)

1199094sw (119896 + 1) = sin (1199084sw (119896) 119905 (119896))

11990641sw (119896) = 119888 (minus31199084sw (119896) +1199081sw (119896) +1199083sw (119896) + sdot sdot sdot

+1199085sw (119896))

(26)

and the fifth firefly1198735 is given by1199085sw (119896 + 1) = 1199085sw (119896) + 11990651sw (119896)

1199095sw (119896 + 1) = sin (1199085sw (119896) 119905 (119896))

11990651sw (119896) = 119888 (minus21199085sw (119896) +1199084sw (119896) +1199081sw (119896))

(27)

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

1050

minus05minus1

xi

0 02 04 06 08 1 212 14 16 18

t(k)

x1x2x3

x4x5

(a)

403020100

wi

0 02 04 06 08 1 212 14 16 18

t(k)

w1

w2

w3

w4

w5

(b)

210

minus1minus2x

1minusxi+1

0 02 04 06 08 1 212 14 16 18

x1 minus x2x1 minus x3

x1 minus x4x1 minus x5

t(k)

(c)

Figure 4 Temporal dynamics of (a) 119909119894(119896) and (b) 119908

119894(119896) with 119894 = 1 2 5 and error (c) 1199091(119896) minus 119909119894+1(119896) with 119894 = 1 2 4 in the uncoupled

nearest-neighbor network

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x2

x1

(a)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x3

x1

(b)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x4

x1

(c)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x5

x1

(d)

Figure 5 Phase diagrams (a) 1199091 versus 1199092 (b) 1199091 versus 1199093 (c) 1199091 versus 1199094 and (d) 1199091 versus 1199095 in the uncoupled nearest-neighbor network

where 119909(119896) is the simple representation of a firefly in discrete-time 119908(119896) = 2120587119891 is the angular frequency in radians with119891 the blink frequency 119905(119896) is the sampling time and 119896 is thenumber of iterations

Isolated firefly (7) does not receive stimulus from anyother firefly in the network and it is flashing with a periodof time of 119905 seconds If a firefly blinks with a certain periodwe take on the sequence of flashes as a circular unitary

Mathematical Problems in Engineering 5

0 02 04 06 08 1 212 14 16 18

t(k)

xi

10

minus1

x1x2x3

x4x5

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

wi

20

40

0

w1

w2

w3

w4

w5

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

x1minusxi+1 1

0minus1

x1 minus x2x1 minus x3

x1 minus x4x1 minus x5

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

w1minuswi+1 20

0minus20minus40

w1 minus w2

w1 minus w3

w1 minus w4

w1 minus w5

(d)

Figure 6 Temporal dynamics of (a) 119909119894(119896) and (b) 119908

119894(119896) with 119894 = 1 2 5 and errors (c) 1199091(119896) minus 119909119894+1(119896) and (d) 1199081(119896) minus 119908119894+1(119896) with

119894 = 1 2 4 in the coupled nearest-neighbor network with 119888 = 001

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x2

x1

(a)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x3

x1

(b)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x4

x1

(c)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x5

x1

(d)

Figure 7 Phase diagrams (a) 1199091 versus 1199092 (b) 1199091 versus 1199093 (c) 1199091 versus 1199094 and (d) 1199091 versus 1199095 in the coupled nearest-neighbor network

motion see Figure 1 where the angular position 120579 is a pointin the unitary circle and a short flash or pulse of the fireflyoccurs when a rotation is completed

Figure 2 shows the behavior of isolated firefly (7) forinitial conditions 119909(0) = 0 119905(0) = 0 119891 = 2Hz and 119896 = 600iterations (note that for appreciation purposes we use theinterpolation option of the Matlab simulation software)

4 Network Synchronization of CoupledFireflies in Different Connection Topologies

In this section network synchronization of coupled fireflies(7) in different topologies and the corresponding numericalresults are reported In the following cases of studywe assumethat all the fireflies are connected without self-loops and

6 Mathematical Problems in Engineering

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 8 Synchronization diagram for the nearest-neighbor net-work

N1N2

N3

N4

N5

Figure 9 Graph for a star network corresponding to A119904

without multiple edges between two fireflies To be consistentwith notation in complex network (1)-(2) we have consideredthe following change of variable

x119894= [

1199091198941

1199091198942] = [

1199081198941

1199091198942] = [

119908119894

119909119894

] (8)

41 Nearest-Neighbor Coupled Network First we considera complex network according to (1)-(2) composed of 119873 =

5 coupled fireflies periodic oscillators (7) as fundamentalnodes in nearest-neighbor coupled topology see Figure 3The corresponding state equations to this dynamical networkare arranged as follows the first firefly1198731 is given by

1199081 (119896 + 1) = 1199081 (119896) + 11990611 (119896)

1199091 (119896 + 1) = sin (1199081 (119896) 119905 (119896))

11990611 (119896) = 119888 (minus21199081 (119896) +1199082 (119896) +1199085 (119896))

(9)

the second firefly1198732 is given by

1199082 (119896 + 1) = 1199082 (119896) + 11990621 (119896)

1199092 (119896 + 1) = sin (1199082 (119896) 119905 (119896))

11990621 (119896) = 119888 (minus21199082 (119896) +1199081 (119896) +1199083 (119896))

(10)

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 10 Synchronization diagram for the star network

N1

N2N3

N4

N5

Figure 11 Graph for a small-world network corresponding to Asw

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 12 Synchronization diagram for the small-world network

the third firefly1198733 is given by

1199083 (119896 + 1) = 1199083 (119896) + 11990631 (119896)

1199093 (119896 + 1) = sin (1199083 (119896) 119905 (119896))

11990631 (119896) = 119888 (minus21199083 (119896) +1199082 (119896) +1199084 (119896))

(11)

Mathematical Problems in Engineering 7

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

xisw

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

40

0wisw

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

x1s

wminusx(i+

1)sw

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

0minus40

w1s

wminusw(i+

1)sw

(d)

Figure 13 Small-world network in mass

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

Figure 14 Graph of outer synchronization between two coupledcomplex networks with different topologies

the fourth firefly1198734 is given by

1199084 (119896 + 1) = 1199084 (119896) + 11990641 (119896)

1199094 (119896 + 1) = sin (1199084 (119896) 119905 (119896))

11990641 (119896) = 119888 (minus21199084 (119896) +1199083 (119896) +1199085 (119896))

(12)

and the fifth firefly1198735 is given by

1199085 (119896 + 1) = 1199085 (119896) + 11990651 (119896)

1199095 (119896 + 1) = sin (1199085 (119896) 119905 (119896))

11990651 (119896) = 119888 (minus21199085 (119896) +1199084 (119896) +1199081 (119896))

(13)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (14)

The corresponding coupling matrixA for the network shownin Figure 3 is given by

A =

[

[

[

[

[

[

[

[

[

minus2 1 0 0 11 minus2 1 0 00 1 minus2 1 00 0 1 minus2 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(15)

For illustrative purposes we present the scenario where theoscillators fireflies of the network are uncoupled (isolated

fireflies) From (9) to (28) by choosing 119888 = 0 the controlinputs 119906

1198941 = 0 therefore the five isolated fireflies of thenetwork are unsynchronized assuming different initial con-ditions Figures 4 and 5 show the collective behavior of fiveisolated fireflies oscillators temporal dynamics of states 119909

119894(119896)

and 119908119894(119896) for 119894 = 1 2 5 synchronization errors 1199091(119896) minus

119909119894+1(119896) for 119894 = 1 2 3 4 and phase diagrams among the

oscillators fireflies 1199091 versus 119909119894 for 119894 = 2 3 4 5 with initialconditions 1199081(0) = 21205872 1199082(0) = 21205875 1199083(0) = 212058711199084(0) = 21205876 1199085(0) = 21205874 1199091(0) = 1199092(0) = 1199093(0) =1199094(0) = 1199095(0) = 0 119905(0) = 0 and 119896 = 600 Considernow the scenario where the oscillators fireflies of the networkare coupled in nearest-neighbor topology (see Figure 3) Fornetwork synchronization purposes we increase the couplingstrength values of 119888 for 119888 gt 0 Let 119888 = 001 with the sameinitial conditions as well as in uncoupled nearest-neighbornetwork Figure 6 shows the temporal dynamics and errorswhere1199081(119896)minus119908119894+1(119896) is computed to observe the behavior ofthe error among the angular frequencies of the synchronizedoscillators

In Figure 7 we can appreciate the phase diagrams whereafter a transient the involved signals overlap in a line of45 degrees denoting synchronization From Figures 6 and 7we can deduce that with this coupling strength and initialconditions the network is synchronizedWe obtained a rangeof coupling strength 119888 numerically We compute (for 119896 =

5000 so the calculation range of 119888 is approximate) a sweepof 119888 from 0 to 1 at intervals of 00001 After removing thefirst 4000 iterations of each of the states we compare the errorbetween states 119909

119894+1(119896) and 1199091(119896) for 119894 = 1 2 3 4 where if thesum of absolute value of the errors is greater than 001 thenwe established that the network is not synchronized (ie after4000 iterations if (|(1199091(119896) minus 1199092(119896))| + |(1199091(119896) minus 1199093(119896))| +|(1199091(119896) minus 1199094(119896))| + |(1199091(119896) minus 1199095(119896))|) gt 001 we establishedthat there is no synchronization in the network) Figure 8shows a diagram that we called synchronization diagramwhere the value of 0 denotes no synchronization and thevalue of 1 denotes synchronization thus we can see the range00019 lt 119888 lt 05521 where the nodes in the network aresynchronized

8 Mathematical Problems in Engineering

10

minus1xixis

0 05 1 15 2 25 3 35

t(k)

xixis

(a)

6040200w

iwis

0 05 1 15 2 25 3 35

t(k)

wi

wis

(b)

20

minus2ximinusxis

0 05 1 15 2 25 3 35

t(k)

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

100

minus10wiminuswis

0 05 1 15 2 25 3 35

t(k)

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 15 Dynamics for two uncoupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and

errors (c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xi xis

(a)

0 05 1 15 2 25 3 35

t(k)

6040200w

iwis

wi wis

(b)

0 05 1 15 2 25 3 35

t(k)

0051

minus05ximinusxis

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

0 05 1 15 2 25 3 35

t(k)

10

0minus10w

iminuswis

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 16 Dynamics for two coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and errors

(c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

42 Star Coupled Network Now we consider a complexnetwork according to (1)-(2) composed of 119873 = 5 coupledfireflies periodic oscillators (7) as fundamental nodes instar coupled topology see Figure 9 The corresponding stateequations to this dynamical network are arranged as followsthe first firefly1198731 is given by

1199081119904 (119896 + 1) = 1199081119904 (119896) + 11990611119904 (119896)

1199091119904 (119896 + 1) = sin (1199081119904 (119896) 119905 (119896))

11990611119904 (119896) = 119888 (minus41199081119904 (119896) +1199082119904 (119896) + sdot sdot sdot +1199083119904 (119896)

+1199084119904 (119896) +1199085119904 (119896))

(16)

the second firefly1198732 is given by

1199082119904 (119896 + 1) = 1199082119904 (119896) + 11990621119904 (119896)

1199092119904 (119896 + 1) = sin (1199082119904 (119896) 119905 (119896))

11990621119904 (119896) = 119888 (minus1199082119904 (119896) +1199081119904 (119896))

(17)

the third firefly1198733 is given by

1199083119904 (119896 + 1) = 1199083119904 (119896) + 11990631119904 (119896)

1199093119904 (119896 + 1) = sin (1199083119904 (119896) 119905 (119896))

11990631119904 (119896) = 119888 (minus1199083119904 (119896) +1199081119904 (119896))

(18)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 17 Synchronization diagram for outer synchronization ofnearest-neighbor-star networks

the fourth firefly1198734 is given by

1199084119904 (119896 + 1) = 1199084119904 (119896) + 11990641119904 (119896)

1199094119904 (119896 + 1) = sin (1199084119904 (119896) 119905 (119896))

11990641119904 (119896) = 119888 (minus1199084119904 (119896) +1199081119904 (119896))

(19)

and the fifth firefly1198735 is given by

1199085119904 (119896 + 1) = 1199085119904 (119896) + 11990651119904 (119896)

1199095119904 (119896 + 1) = sin (1199085119904 (119896) 119905 (119896))

11990651119904 (119896) = 119888 (minus1199085119904 (119896) +1199081119904 (119896))

(20)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (21)

The corresponding coupling matrix A119904 for the networkshown in Figure 9 is given by

A119904 =

[

[

[

[

[

[

[

[

[

minus4 1 1 1 11 minus1 0 0 01 0 minus1 0 01 0 0 minus1 01 0 0 0 minus1

]

]

]

]

]

]

]

]

]

(22)

where subscript 119904 refers to the star network We carry out thenumerical simulations in the star network with 119888 = 001 andinitial conditions 1199081119904(0) = 212058725 1199082119904(0) = 212058755 1199083119904(0) =212058715 1199084119904(0) = 212058765 1199085119904(0) = 212058745 1199091119904(0) = 1199092119904(0) =1199093119904(0) = 1199094119904(0) = 1199095119904(0) = 0 119905(0) = 0 and 119896 = 600 Takinginto account the above conditions we obtain similar results asin the nearest-neighbor network and synchronization amongthe nodes was achieved Figure 10 shows the synchronizationdiagram where we can see the range 00021 lt 119888 lt 03995where the nodes in the network are synchronized

43 Small-World Coupled Network A network with complexcoupled topology can be represented by a random graph butour intuition tells us clearly that many real complex networksare neither completely randomnor completely regular Some-thing that we must emphasize is that most of the biologicaltechnological and social networks lie between these twoextremesThese systems are highly clustered as regular arrayshowever they have a characteristic path length like randomnetworks see [21] These types of networks are called small-world networks in analogy to the small-world phenomenon(popularly known as six degrees of separation)

In general terms these networks are regular arrange-ments made by connections or reconnections of pairs ofrandomly chosen nodes Even a very small number ofthese added connections commonly called shortcuts do notchange the local properties (very high clustering typical of aregular network) and cause those typical random networksvalues to be present in the average path length see [22]

Now we consider a complex network according to (1)-(2) composed of 119873 = 5 coupled fireflies periodic oscillators(7) as fundamental nodes in small-world coupled topologysee Figure 11 The corresponding state equations to thisdynamical network are arranged as follows the first firefly1198731is given by1199081sw (119896 + 1) = 1199081sw (119896) + 11990611sw (119896)

1199091sw (119896 + 1) = sin (1199081sw (119896) 119905 (119896))

11990611sw (119896) = 119888 (minus31199081sw (119896) +1199082sw (119896) +1199084sw (119896) + sdot sdot sdot

+1199085sw (119896))

(23)

the second firefly1198732 is given by1199082sw (119896 + 1) = 1199082sw (119896) + 11990621sw (119896)

1199092sw (119896 + 1) = sin (1199082sw (119896) 119905 (119896))

11990621sw (119896) = 119888 (minus21199082sw (119896) +1199081sw (119896) +1199083sw (119896))

(24)

the third firefly1198733 is given by1199083sw (119896 + 1) = 1199083sw (119896) + 11990631sw (119896)

1199093sw (119896 + 1) = sin (1199083sw (119896) 119905 (119896))

11990631sw (119896) = 119888 (minus21199083sw (119896) +1199082sw (119896) +1199084sw (119896))

(25)

the fourth firefly1198734 is given by1199084sw (119896 + 1) = 1199084sw (119896) + 11990641sw (119896)

1199094sw (119896 + 1) = sin (1199084sw (119896) 119905 (119896))

11990641sw (119896) = 119888 (minus31199084sw (119896) +1199081sw (119896) +1199083sw (119896) + sdot sdot sdot

+1199085sw (119896))

(26)

and the fifth firefly1198735 is given by1199085sw (119896 + 1) = 1199085sw (119896) + 11990651sw (119896)

1199095sw (119896 + 1) = sin (1199085sw (119896) 119905 (119896))

11990651sw (119896) = 119888 (minus21199085sw (119896) +1199084sw (119896) +1199081sw (119896))

(27)

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 02 04 06 08 1 212 14 16 18

t(k)

xi

10

minus1

x1x2x3

x4x5

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

wi

20

40

0

w1

w2

w3

w4

w5

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

x1minusxi+1 1

0minus1

x1 minus x2x1 minus x3

x1 minus x4x1 minus x5

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

w1minuswi+1 20

0minus20minus40

w1 minus w2

w1 minus w3

w1 minus w4

w1 minus w5

(d)

Figure 6 Temporal dynamics of (a) 119909119894(119896) and (b) 119908

119894(119896) with 119894 = 1 2 5 and errors (c) 1199091(119896) minus 119909119894+1(119896) and (d) 1199081(119896) minus 119908119894+1(119896) with

119894 = 1 2 4 in the coupled nearest-neighbor network with 119888 = 001

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x2

x1

(a)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x3

x1

(b)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x4

x1

(c)

1

05

0

minus05

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x5

x1

(d)

Figure 7 Phase diagrams (a) 1199091 versus 1199092 (b) 1199091 versus 1199093 (c) 1199091 versus 1199094 and (d) 1199091 versus 1199095 in the coupled nearest-neighbor network

motion see Figure 1 where the angular position 120579 is a pointin the unitary circle and a short flash or pulse of the fireflyoccurs when a rotation is completed

Figure 2 shows the behavior of isolated firefly (7) forinitial conditions 119909(0) = 0 119905(0) = 0 119891 = 2Hz and 119896 = 600iterations (note that for appreciation purposes we use theinterpolation option of the Matlab simulation software)

4 Network Synchronization of CoupledFireflies in Different Connection Topologies

In this section network synchronization of coupled fireflies(7) in different topologies and the corresponding numericalresults are reported In the following cases of studywe assumethat all the fireflies are connected without self-loops and

6 Mathematical Problems in Engineering

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 8 Synchronization diagram for the nearest-neighbor net-work

N1N2

N3

N4

N5

Figure 9 Graph for a star network corresponding to A119904

without multiple edges between two fireflies To be consistentwith notation in complex network (1)-(2) we have consideredthe following change of variable

x119894= [

1199091198941

1199091198942] = [

1199081198941

1199091198942] = [

119908119894

119909119894

] (8)

41 Nearest-Neighbor Coupled Network First we considera complex network according to (1)-(2) composed of 119873 =

5 coupled fireflies periodic oscillators (7) as fundamentalnodes in nearest-neighbor coupled topology see Figure 3The corresponding state equations to this dynamical networkare arranged as follows the first firefly1198731 is given by

1199081 (119896 + 1) = 1199081 (119896) + 11990611 (119896)

1199091 (119896 + 1) = sin (1199081 (119896) 119905 (119896))

11990611 (119896) = 119888 (minus21199081 (119896) +1199082 (119896) +1199085 (119896))

(9)

the second firefly1198732 is given by

1199082 (119896 + 1) = 1199082 (119896) + 11990621 (119896)

1199092 (119896 + 1) = sin (1199082 (119896) 119905 (119896))

11990621 (119896) = 119888 (minus21199082 (119896) +1199081 (119896) +1199083 (119896))

(10)

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 10 Synchronization diagram for the star network

N1

N2N3

N4

N5

Figure 11 Graph for a small-world network corresponding to Asw

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 12 Synchronization diagram for the small-world network

the third firefly1198733 is given by

1199083 (119896 + 1) = 1199083 (119896) + 11990631 (119896)

1199093 (119896 + 1) = sin (1199083 (119896) 119905 (119896))

11990631 (119896) = 119888 (minus21199083 (119896) +1199082 (119896) +1199084 (119896))

(11)

Mathematical Problems in Engineering 7

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

xisw

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

40

0wisw

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

x1s

wminusx(i+

1)sw

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

0minus40

w1s

wminusw(i+

1)sw

(d)

Figure 13 Small-world network in mass

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

Figure 14 Graph of outer synchronization between two coupledcomplex networks with different topologies

the fourth firefly1198734 is given by

1199084 (119896 + 1) = 1199084 (119896) + 11990641 (119896)

1199094 (119896 + 1) = sin (1199084 (119896) 119905 (119896))

11990641 (119896) = 119888 (minus21199084 (119896) +1199083 (119896) +1199085 (119896))

(12)

and the fifth firefly1198735 is given by

1199085 (119896 + 1) = 1199085 (119896) + 11990651 (119896)

1199095 (119896 + 1) = sin (1199085 (119896) 119905 (119896))

11990651 (119896) = 119888 (minus21199085 (119896) +1199084 (119896) +1199081 (119896))

(13)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (14)

The corresponding coupling matrixA for the network shownin Figure 3 is given by

A =

[

[

[

[

[

[

[

[

[

minus2 1 0 0 11 minus2 1 0 00 1 minus2 1 00 0 1 minus2 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(15)

For illustrative purposes we present the scenario where theoscillators fireflies of the network are uncoupled (isolated

fireflies) From (9) to (28) by choosing 119888 = 0 the controlinputs 119906

1198941 = 0 therefore the five isolated fireflies of thenetwork are unsynchronized assuming different initial con-ditions Figures 4 and 5 show the collective behavior of fiveisolated fireflies oscillators temporal dynamics of states 119909

119894(119896)

and 119908119894(119896) for 119894 = 1 2 5 synchronization errors 1199091(119896) minus

119909119894+1(119896) for 119894 = 1 2 3 4 and phase diagrams among the

oscillators fireflies 1199091 versus 119909119894 for 119894 = 2 3 4 5 with initialconditions 1199081(0) = 21205872 1199082(0) = 21205875 1199083(0) = 212058711199084(0) = 21205876 1199085(0) = 21205874 1199091(0) = 1199092(0) = 1199093(0) =1199094(0) = 1199095(0) = 0 119905(0) = 0 and 119896 = 600 Considernow the scenario where the oscillators fireflies of the networkare coupled in nearest-neighbor topology (see Figure 3) Fornetwork synchronization purposes we increase the couplingstrength values of 119888 for 119888 gt 0 Let 119888 = 001 with the sameinitial conditions as well as in uncoupled nearest-neighbornetwork Figure 6 shows the temporal dynamics and errorswhere1199081(119896)minus119908119894+1(119896) is computed to observe the behavior ofthe error among the angular frequencies of the synchronizedoscillators

In Figure 7 we can appreciate the phase diagrams whereafter a transient the involved signals overlap in a line of45 degrees denoting synchronization From Figures 6 and 7we can deduce that with this coupling strength and initialconditions the network is synchronizedWe obtained a rangeof coupling strength 119888 numerically We compute (for 119896 =

5000 so the calculation range of 119888 is approximate) a sweepof 119888 from 0 to 1 at intervals of 00001 After removing thefirst 4000 iterations of each of the states we compare the errorbetween states 119909

119894+1(119896) and 1199091(119896) for 119894 = 1 2 3 4 where if thesum of absolute value of the errors is greater than 001 thenwe established that the network is not synchronized (ie after4000 iterations if (|(1199091(119896) minus 1199092(119896))| + |(1199091(119896) minus 1199093(119896))| +|(1199091(119896) minus 1199094(119896))| + |(1199091(119896) minus 1199095(119896))|) gt 001 we establishedthat there is no synchronization in the network) Figure 8shows a diagram that we called synchronization diagramwhere the value of 0 denotes no synchronization and thevalue of 1 denotes synchronization thus we can see the range00019 lt 119888 lt 05521 where the nodes in the network aresynchronized

8 Mathematical Problems in Engineering

10

minus1xixis

0 05 1 15 2 25 3 35

t(k)

xixis

(a)

6040200w

iwis

0 05 1 15 2 25 3 35

t(k)

wi

wis

(b)

20

minus2ximinusxis

0 05 1 15 2 25 3 35

t(k)

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

100

minus10wiminuswis

0 05 1 15 2 25 3 35

t(k)

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 15 Dynamics for two uncoupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and

errors (c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xi xis

(a)

0 05 1 15 2 25 3 35

t(k)

6040200w

iwis

wi wis

(b)

0 05 1 15 2 25 3 35

t(k)

0051

minus05ximinusxis

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

0 05 1 15 2 25 3 35

t(k)

10

0minus10w

iminuswis

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 16 Dynamics for two coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and errors

(c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

42 Star Coupled Network Now we consider a complexnetwork according to (1)-(2) composed of 119873 = 5 coupledfireflies periodic oscillators (7) as fundamental nodes instar coupled topology see Figure 9 The corresponding stateequations to this dynamical network are arranged as followsthe first firefly1198731 is given by

1199081119904 (119896 + 1) = 1199081119904 (119896) + 11990611119904 (119896)

1199091119904 (119896 + 1) = sin (1199081119904 (119896) 119905 (119896))

11990611119904 (119896) = 119888 (minus41199081119904 (119896) +1199082119904 (119896) + sdot sdot sdot +1199083119904 (119896)

+1199084119904 (119896) +1199085119904 (119896))

(16)

the second firefly1198732 is given by

1199082119904 (119896 + 1) = 1199082119904 (119896) + 11990621119904 (119896)

1199092119904 (119896 + 1) = sin (1199082119904 (119896) 119905 (119896))

11990621119904 (119896) = 119888 (minus1199082119904 (119896) +1199081119904 (119896))

(17)

the third firefly1198733 is given by

1199083119904 (119896 + 1) = 1199083119904 (119896) + 11990631119904 (119896)

1199093119904 (119896 + 1) = sin (1199083119904 (119896) 119905 (119896))

11990631119904 (119896) = 119888 (minus1199083119904 (119896) +1199081119904 (119896))

(18)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 17 Synchronization diagram for outer synchronization ofnearest-neighbor-star networks

the fourth firefly1198734 is given by

1199084119904 (119896 + 1) = 1199084119904 (119896) + 11990641119904 (119896)

1199094119904 (119896 + 1) = sin (1199084119904 (119896) 119905 (119896))

11990641119904 (119896) = 119888 (minus1199084119904 (119896) +1199081119904 (119896))

(19)

and the fifth firefly1198735 is given by

1199085119904 (119896 + 1) = 1199085119904 (119896) + 11990651119904 (119896)

1199095119904 (119896 + 1) = sin (1199085119904 (119896) 119905 (119896))

11990651119904 (119896) = 119888 (minus1199085119904 (119896) +1199081119904 (119896))

(20)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (21)

The corresponding coupling matrix A119904 for the networkshown in Figure 9 is given by

A119904 =

[

[

[

[

[

[

[

[

[

minus4 1 1 1 11 minus1 0 0 01 0 minus1 0 01 0 0 minus1 01 0 0 0 minus1

]

]

]

]

]

]

]

]

]

(22)

where subscript 119904 refers to the star network We carry out thenumerical simulations in the star network with 119888 = 001 andinitial conditions 1199081119904(0) = 212058725 1199082119904(0) = 212058755 1199083119904(0) =212058715 1199084119904(0) = 212058765 1199085119904(0) = 212058745 1199091119904(0) = 1199092119904(0) =1199093119904(0) = 1199094119904(0) = 1199095119904(0) = 0 119905(0) = 0 and 119896 = 600 Takinginto account the above conditions we obtain similar results asin the nearest-neighbor network and synchronization amongthe nodes was achieved Figure 10 shows the synchronizationdiagram where we can see the range 00021 lt 119888 lt 03995where the nodes in the network are synchronized

43 Small-World Coupled Network A network with complexcoupled topology can be represented by a random graph butour intuition tells us clearly that many real complex networksare neither completely randomnor completely regular Some-thing that we must emphasize is that most of the biologicaltechnological and social networks lie between these twoextremesThese systems are highly clustered as regular arrayshowever they have a characteristic path length like randomnetworks see [21] These types of networks are called small-world networks in analogy to the small-world phenomenon(popularly known as six degrees of separation)

In general terms these networks are regular arrange-ments made by connections or reconnections of pairs ofrandomly chosen nodes Even a very small number ofthese added connections commonly called shortcuts do notchange the local properties (very high clustering typical of aregular network) and cause those typical random networksvalues to be present in the average path length see [22]

Now we consider a complex network according to (1)-(2) composed of 119873 = 5 coupled fireflies periodic oscillators(7) as fundamental nodes in small-world coupled topologysee Figure 11 The corresponding state equations to thisdynamical network are arranged as follows the first firefly1198731is given by1199081sw (119896 + 1) = 1199081sw (119896) + 11990611sw (119896)

1199091sw (119896 + 1) = sin (1199081sw (119896) 119905 (119896))

11990611sw (119896) = 119888 (minus31199081sw (119896) +1199082sw (119896) +1199084sw (119896) + sdot sdot sdot

+1199085sw (119896))

(23)

the second firefly1198732 is given by1199082sw (119896 + 1) = 1199082sw (119896) + 11990621sw (119896)

1199092sw (119896 + 1) = sin (1199082sw (119896) 119905 (119896))

11990621sw (119896) = 119888 (minus21199082sw (119896) +1199081sw (119896) +1199083sw (119896))

(24)

the third firefly1198733 is given by1199083sw (119896 + 1) = 1199083sw (119896) + 11990631sw (119896)

1199093sw (119896 + 1) = sin (1199083sw (119896) 119905 (119896))

11990631sw (119896) = 119888 (minus21199083sw (119896) +1199082sw (119896) +1199084sw (119896))

(25)

the fourth firefly1198734 is given by1199084sw (119896 + 1) = 1199084sw (119896) + 11990641sw (119896)

1199094sw (119896 + 1) = sin (1199084sw (119896) 119905 (119896))

11990641sw (119896) = 119888 (minus31199084sw (119896) +1199081sw (119896) +1199083sw (119896) + sdot sdot sdot

+1199085sw (119896))

(26)

and the fifth firefly1198735 is given by1199085sw (119896 + 1) = 1199085sw (119896) + 11990651sw (119896)

1199095sw (119896 + 1) = sin (1199085sw (119896) 119905 (119896))

11990651sw (119896) = 119888 (minus21199085sw (119896) +1199084sw (119896) +1199081sw (119896))

(27)

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 8 Synchronization diagram for the nearest-neighbor net-work

N1N2

N3

N4

N5

Figure 9 Graph for a star network corresponding to A119904

without multiple edges between two fireflies To be consistentwith notation in complex network (1)-(2) we have consideredthe following change of variable

x119894= [

1199091198941

1199091198942] = [

1199081198941

1199091198942] = [

119908119894

119909119894

] (8)

41 Nearest-Neighbor Coupled Network First we considera complex network according to (1)-(2) composed of 119873 =

5 coupled fireflies periodic oscillators (7) as fundamentalnodes in nearest-neighbor coupled topology see Figure 3The corresponding state equations to this dynamical networkare arranged as follows the first firefly1198731 is given by

1199081 (119896 + 1) = 1199081 (119896) + 11990611 (119896)

1199091 (119896 + 1) = sin (1199081 (119896) 119905 (119896))

11990611 (119896) = 119888 (minus21199081 (119896) +1199082 (119896) +1199085 (119896))

(9)

the second firefly1198732 is given by

1199082 (119896 + 1) = 1199082 (119896) + 11990621 (119896)

1199092 (119896 + 1) = sin (1199082 (119896) 119905 (119896))

11990621 (119896) = 119888 (minus21199082 (119896) +1199081 (119896) +1199083 (119896))

(10)

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 10 Synchronization diagram for the star network

N1

N2N3

N4

N5

Figure 11 Graph for a small-world network corresponding to Asw

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 12 Synchronization diagram for the small-world network

the third firefly1198733 is given by

1199083 (119896 + 1) = 1199083 (119896) + 11990631 (119896)

1199093 (119896 + 1) = sin (1199083 (119896) 119905 (119896))

11990631 (119896) = 119888 (minus21199083 (119896) +1199082 (119896) +1199084 (119896))

(11)

Mathematical Problems in Engineering 7

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

xisw

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

40

0wisw

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

x1s

wminusx(i+

1)sw

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

0minus40

w1s

wminusw(i+

1)sw

(d)

Figure 13 Small-world network in mass

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

Figure 14 Graph of outer synchronization between two coupledcomplex networks with different topologies

the fourth firefly1198734 is given by

1199084 (119896 + 1) = 1199084 (119896) + 11990641 (119896)

1199094 (119896 + 1) = sin (1199084 (119896) 119905 (119896))

11990641 (119896) = 119888 (minus21199084 (119896) +1199083 (119896) +1199085 (119896))

(12)

and the fifth firefly1198735 is given by

1199085 (119896 + 1) = 1199085 (119896) + 11990651 (119896)

1199095 (119896 + 1) = sin (1199085 (119896) 119905 (119896))

11990651 (119896) = 119888 (minus21199085 (119896) +1199084 (119896) +1199081 (119896))

(13)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (14)

The corresponding coupling matrixA for the network shownin Figure 3 is given by

A =

[

[

[

[

[

[

[

[

[

minus2 1 0 0 11 minus2 1 0 00 1 minus2 1 00 0 1 minus2 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(15)

For illustrative purposes we present the scenario where theoscillators fireflies of the network are uncoupled (isolated

fireflies) From (9) to (28) by choosing 119888 = 0 the controlinputs 119906

1198941 = 0 therefore the five isolated fireflies of thenetwork are unsynchronized assuming different initial con-ditions Figures 4 and 5 show the collective behavior of fiveisolated fireflies oscillators temporal dynamics of states 119909

119894(119896)

and 119908119894(119896) for 119894 = 1 2 5 synchronization errors 1199091(119896) minus

119909119894+1(119896) for 119894 = 1 2 3 4 and phase diagrams among the

oscillators fireflies 1199091 versus 119909119894 for 119894 = 2 3 4 5 with initialconditions 1199081(0) = 21205872 1199082(0) = 21205875 1199083(0) = 212058711199084(0) = 21205876 1199085(0) = 21205874 1199091(0) = 1199092(0) = 1199093(0) =1199094(0) = 1199095(0) = 0 119905(0) = 0 and 119896 = 600 Considernow the scenario where the oscillators fireflies of the networkare coupled in nearest-neighbor topology (see Figure 3) Fornetwork synchronization purposes we increase the couplingstrength values of 119888 for 119888 gt 0 Let 119888 = 001 with the sameinitial conditions as well as in uncoupled nearest-neighbornetwork Figure 6 shows the temporal dynamics and errorswhere1199081(119896)minus119908119894+1(119896) is computed to observe the behavior ofthe error among the angular frequencies of the synchronizedoscillators

In Figure 7 we can appreciate the phase diagrams whereafter a transient the involved signals overlap in a line of45 degrees denoting synchronization From Figures 6 and 7we can deduce that with this coupling strength and initialconditions the network is synchronizedWe obtained a rangeof coupling strength 119888 numerically We compute (for 119896 =

5000 so the calculation range of 119888 is approximate) a sweepof 119888 from 0 to 1 at intervals of 00001 After removing thefirst 4000 iterations of each of the states we compare the errorbetween states 119909

119894+1(119896) and 1199091(119896) for 119894 = 1 2 3 4 where if thesum of absolute value of the errors is greater than 001 thenwe established that the network is not synchronized (ie after4000 iterations if (|(1199091(119896) minus 1199092(119896))| + |(1199091(119896) minus 1199093(119896))| +|(1199091(119896) minus 1199094(119896))| + |(1199091(119896) minus 1199095(119896))|) gt 001 we establishedthat there is no synchronization in the network) Figure 8shows a diagram that we called synchronization diagramwhere the value of 0 denotes no synchronization and thevalue of 1 denotes synchronization thus we can see the range00019 lt 119888 lt 05521 where the nodes in the network aresynchronized

8 Mathematical Problems in Engineering

10

minus1xixis

0 05 1 15 2 25 3 35

t(k)

xixis

(a)

6040200w

iwis

0 05 1 15 2 25 3 35

t(k)

wi

wis

(b)

20

minus2ximinusxis

0 05 1 15 2 25 3 35

t(k)

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

100

minus10wiminuswis

0 05 1 15 2 25 3 35

t(k)

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 15 Dynamics for two uncoupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and

errors (c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xi xis

(a)

0 05 1 15 2 25 3 35

t(k)

6040200w

iwis

wi wis

(b)

0 05 1 15 2 25 3 35

t(k)

0051

minus05ximinusxis

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

0 05 1 15 2 25 3 35

t(k)

10

0minus10w

iminuswis

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 16 Dynamics for two coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and errors

(c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

42 Star Coupled Network Now we consider a complexnetwork according to (1)-(2) composed of 119873 = 5 coupledfireflies periodic oscillators (7) as fundamental nodes instar coupled topology see Figure 9 The corresponding stateequations to this dynamical network are arranged as followsthe first firefly1198731 is given by

1199081119904 (119896 + 1) = 1199081119904 (119896) + 11990611119904 (119896)

1199091119904 (119896 + 1) = sin (1199081119904 (119896) 119905 (119896))

11990611119904 (119896) = 119888 (minus41199081119904 (119896) +1199082119904 (119896) + sdot sdot sdot +1199083119904 (119896)

+1199084119904 (119896) +1199085119904 (119896))

(16)

the second firefly1198732 is given by

1199082119904 (119896 + 1) = 1199082119904 (119896) + 11990621119904 (119896)

1199092119904 (119896 + 1) = sin (1199082119904 (119896) 119905 (119896))

11990621119904 (119896) = 119888 (minus1199082119904 (119896) +1199081119904 (119896))

(17)

the third firefly1198733 is given by

1199083119904 (119896 + 1) = 1199083119904 (119896) + 11990631119904 (119896)

1199093119904 (119896 + 1) = sin (1199083119904 (119896) 119905 (119896))

11990631119904 (119896) = 119888 (minus1199083119904 (119896) +1199081119904 (119896))

(18)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 17 Synchronization diagram for outer synchronization ofnearest-neighbor-star networks

the fourth firefly1198734 is given by

1199084119904 (119896 + 1) = 1199084119904 (119896) + 11990641119904 (119896)

1199094119904 (119896 + 1) = sin (1199084119904 (119896) 119905 (119896))

11990641119904 (119896) = 119888 (minus1199084119904 (119896) +1199081119904 (119896))

(19)

and the fifth firefly1198735 is given by

1199085119904 (119896 + 1) = 1199085119904 (119896) + 11990651119904 (119896)

1199095119904 (119896 + 1) = sin (1199085119904 (119896) 119905 (119896))

11990651119904 (119896) = 119888 (minus1199085119904 (119896) +1199081119904 (119896))

(20)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (21)

The corresponding coupling matrix A119904 for the networkshown in Figure 9 is given by

A119904 =

[

[

[

[

[

[

[

[

[

minus4 1 1 1 11 minus1 0 0 01 0 minus1 0 01 0 0 minus1 01 0 0 0 minus1

]

]

]

]

]

]

]

]

]

(22)

where subscript 119904 refers to the star network We carry out thenumerical simulations in the star network with 119888 = 001 andinitial conditions 1199081119904(0) = 212058725 1199082119904(0) = 212058755 1199083119904(0) =212058715 1199084119904(0) = 212058765 1199085119904(0) = 212058745 1199091119904(0) = 1199092119904(0) =1199093119904(0) = 1199094119904(0) = 1199095119904(0) = 0 119905(0) = 0 and 119896 = 600 Takinginto account the above conditions we obtain similar results asin the nearest-neighbor network and synchronization amongthe nodes was achieved Figure 10 shows the synchronizationdiagram where we can see the range 00021 lt 119888 lt 03995where the nodes in the network are synchronized

43 Small-World Coupled Network A network with complexcoupled topology can be represented by a random graph butour intuition tells us clearly that many real complex networksare neither completely randomnor completely regular Some-thing that we must emphasize is that most of the biologicaltechnological and social networks lie between these twoextremesThese systems are highly clustered as regular arrayshowever they have a characteristic path length like randomnetworks see [21] These types of networks are called small-world networks in analogy to the small-world phenomenon(popularly known as six degrees of separation)

In general terms these networks are regular arrange-ments made by connections or reconnections of pairs ofrandomly chosen nodes Even a very small number ofthese added connections commonly called shortcuts do notchange the local properties (very high clustering typical of aregular network) and cause those typical random networksvalues to be present in the average path length see [22]

Now we consider a complex network according to (1)-(2) composed of 119873 = 5 coupled fireflies periodic oscillators(7) as fundamental nodes in small-world coupled topologysee Figure 11 The corresponding state equations to thisdynamical network are arranged as follows the first firefly1198731is given by1199081sw (119896 + 1) = 1199081sw (119896) + 11990611sw (119896)

1199091sw (119896 + 1) = sin (1199081sw (119896) 119905 (119896))

11990611sw (119896) = 119888 (minus31199081sw (119896) +1199082sw (119896) +1199084sw (119896) + sdot sdot sdot

+1199085sw (119896))

(23)

the second firefly1198732 is given by1199082sw (119896 + 1) = 1199082sw (119896) + 11990621sw (119896)

1199092sw (119896 + 1) = sin (1199082sw (119896) 119905 (119896))

11990621sw (119896) = 119888 (minus21199082sw (119896) +1199081sw (119896) +1199083sw (119896))

(24)

the third firefly1198733 is given by1199083sw (119896 + 1) = 1199083sw (119896) + 11990631sw (119896)

1199093sw (119896 + 1) = sin (1199083sw (119896) 119905 (119896))

11990631sw (119896) = 119888 (minus21199083sw (119896) +1199082sw (119896) +1199084sw (119896))

(25)

the fourth firefly1198734 is given by1199084sw (119896 + 1) = 1199084sw (119896) + 11990641sw (119896)

1199094sw (119896 + 1) = sin (1199084sw (119896) 119905 (119896))

11990641sw (119896) = 119888 (minus31199084sw (119896) +1199081sw (119896) +1199083sw (119896) + sdot sdot sdot

+1199085sw (119896))

(26)

and the fifth firefly1198735 is given by1199085sw (119896 + 1) = 1199085sw (119896) + 11990651sw (119896)

1199095sw (119896 + 1) = sin (1199085sw (119896) 119905 (119896))

11990651sw (119896) = 119888 (minus21199085sw (119896) +1199084sw (119896) +1199081sw (119896))

(27)

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

xisw

(a)

0 02 04 06 08 1 212 14 16 18

t(k)

40

0wisw

(b)

0 02 04 06 08 1 212 14 16 18

t(k)

10

minus1

x1s

wminusx(i+

1)sw

(c)

0 02 04 06 08 1 212 14 16 18

t(k)

0minus40

w1s

wminusw(i+

1)sw

(d)

Figure 13 Small-world network in mass

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

Figure 14 Graph of outer synchronization between two coupledcomplex networks with different topologies

the fourth firefly1198734 is given by

1199084 (119896 + 1) = 1199084 (119896) + 11990641 (119896)

1199094 (119896 + 1) = sin (1199084 (119896) 119905 (119896))

11990641 (119896) = 119888 (minus21199084 (119896) +1199083 (119896) +1199085 (119896))

(12)

and the fifth firefly1198735 is given by

1199085 (119896 + 1) = 1199085 (119896) + 11990651 (119896)

1199095 (119896 + 1) = sin (1199085 (119896) 119905 (119896))

11990651 (119896) = 119888 (minus21199085 (119896) +1199084 (119896) +1199081 (119896))

(13)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (14)

The corresponding coupling matrixA for the network shownin Figure 3 is given by

A =

[

[

[

[

[

[

[

[

[

minus2 1 0 0 11 minus2 1 0 00 1 minus2 1 00 0 1 minus2 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(15)

For illustrative purposes we present the scenario where theoscillators fireflies of the network are uncoupled (isolated

fireflies) From (9) to (28) by choosing 119888 = 0 the controlinputs 119906

1198941 = 0 therefore the five isolated fireflies of thenetwork are unsynchronized assuming different initial con-ditions Figures 4 and 5 show the collective behavior of fiveisolated fireflies oscillators temporal dynamics of states 119909

119894(119896)

and 119908119894(119896) for 119894 = 1 2 5 synchronization errors 1199091(119896) minus

119909119894+1(119896) for 119894 = 1 2 3 4 and phase diagrams among the

oscillators fireflies 1199091 versus 119909119894 for 119894 = 2 3 4 5 with initialconditions 1199081(0) = 21205872 1199082(0) = 21205875 1199083(0) = 212058711199084(0) = 21205876 1199085(0) = 21205874 1199091(0) = 1199092(0) = 1199093(0) =1199094(0) = 1199095(0) = 0 119905(0) = 0 and 119896 = 600 Considernow the scenario where the oscillators fireflies of the networkare coupled in nearest-neighbor topology (see Figure 3) Fornetwork synchronization purposes we increase the couplingstrength values of 119888 for 119888 gt 0 Let 119888 = 001 with the sameinitial conditions as well as in uncoupled nearest-neighbornetwork Figure 6 shows the temporal dynamics and errorswhere1199081(119896)minus119908119894+1(119896) is computed to observe the behavior ofthe error among the angular frequencies of the synchronizedoscillators

In Figure 7 we can appreciate the phase diagrams whereafter a transient the involved signals overlap in a line of45 degrees denoting synchronization From Figures 6 and 7we can deduce that with this coupling strength and initialconditions the network is synchronizedWe obtained a rangeof coupling strength 119888 numerically We compute (for 119896 =

5000 so the calculation range of 119888 is approximate) a sweepof 119888 from 0 to 1 at intervals of 00001 After removing thefirst 4000 iterations of each of the states we compare the errorbetween states 119909

119894+1(119896) and 1199091(119896) for 119894 = 1 2 3 4 where if thesum of absolute value of the errors is greater than 001 thenwe established that the network is not synchronized (ie after4000 iterations if (|(1199091(119896) minus 1199092(119896))| + |(1199091(119896) minus 1199093(119896))| +|(1199091(119896) minus 1199094(119896))| + |(1199091(119896) minus 1199095(119896))|) gt 001 we establishedthat there is no synchronization in the network) Figure 8shows a diagram that we called synchronization diagramwhere the value of 0 denotes no synchronization and thevalue of 1 denotes synchronization thus we can see the range00019 lt 119888 lt 05521 where the nodes in the network aresynchronized

8 Mathematical Problems in Engineering

10

minus1xixis

0 05 1 15 2 25 3 35

t(k)

xixis

(a)

6040200w

iwis

0 05 1 15 2 25 3 35

t(k)

wi

wis

(b)

20

minus2ximinusxis

0 05 1 15 2 25 3 35

t(k)

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

100

minus10wiminuswis

0 05 1 15 2 25 3 35

t(k)

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 15 Dynamics for two uncoupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and

errors (c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xi xis

(a)

0 05 1 15 2 25 3 35

t(k)

6040200w

iwis

wi wis

(b)

0 05 1 15 2 25 3 35

t(k)

0051

minus05ximinusxis

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

0 05 1 15 2 25 3 35

t(k)

10

0minus10w

iminuswis

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 16 Dynamics for two coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and errors

(c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

42 Star Coupled Network Now we consider a complexnetwork according to (1)-(2) composed of 119873 = 5 coupledfireflies periodic oscillators (7) as fundamental nodes instar coupled topology see Figure 9 The corresponding stateequations to this dynamical network are arranged as followsthe first firefly1198731 is given by

1199081119904 (119896 + 1) = 1199081119904 (119896) + 11990611119904 (119896)

1199091119904 (119896 + 1) = sin (1199081119904 (119896) 119905 (119896))

11990611119904 (119896) = 119888 (minus41199081119904 (119896) +1199082119904 (119896) + sdot sdot sdot +1199083119904 (119896)

+1199084119904 (119896) +1199085119904 (119896))

(16)

the second firefly1198732 is given by

1199082119904 (119896 + 1) = 1199082119904 (119896) + 11990621119904 (119896)

1199092119904 (119896 + 1) = sin (1199082119904 (119896) 119905 (119896))

11990621119904 (119896) = 119888 (minus1199082119904 (119896) +1199081119904 (119896))

(17)

the third firefly1198733 is given by

1199083119904 (119896 + 1) = 1199083119904 (119896) + 11990631119904 (119896)

1199093119904 (119896 + 1) = sin (1199083119904 (119896) 119905 (119896))

11990631119904 (119896) = 119888 (minus1199083119904 (119896) +1199081119904 (119896))

(18)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 17 Synchronization diagram for outer synchronization ofnearest-neighbor-star networks

the fourth firefly1198734 is given by

1199084119904 (119896 + 1) = 1199084119904 (119896) + 11990641119904 (119896)

1199094119904 (119896 + 1) = sin (1199084119904 (119896) 119905 (119896))

11990641119904 (119896) = 119888 (minus1199084119904 (119896) +1199081119904 (119896))

(19)

and the fifth firefly1198735 is given by

1199085119904 (119896 + 1) = 1199085119904 (119896) + 11990651119904 (119896)

1199095119904 (119896 + 1) = sin (1199085119904 (119896) 119905 (119896))

11990651119904 (119896) = 119888 (minus1199085119904 (119896) +1199081119904 (119896))

(20)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (21)

The corresponding coupling matrix A119904 for the networkshown in Figure 9 is given by

A119904 =

[

[

[

[

[

[

[

[

[

minus4 1 1 1 11 minus1 0 0 01 0 minus1 0 01 0 0 minus1 01 0 0 0 minus1

]

]

]

]

]

]

]

]

]

(22)

where subscript 119904 refers to the star network We carry out thenumerical simulations in the star network with 119888 = 001 andinitial conditions 1199081119904(0) = 212058725 1199082119904(0) = 212058755 1199083119904(0) =212058715 1199084119904(0) = 212058765 1199085119904(0) = 212058745 1199091119904(0) = 1199092119904(0) =1199093119904(0) = 1199094119904(0) = 1199095119904(0) = 0 119905(0) = 0 and 119896 = 600 Takinginto account the above conditions we obtain similar results asin the nearest-neighbor network and synchronization amongthe nodes was achieved Figure 10 shows the synchronizationdiagram where we can see the range 00021 lt 119888 lt 03995where the nodes in the network are synchronized

43 Small-World Coupled Network A network with complexcoupled topology can be represented by a random graph butour intuition tells us clearly that many real complex networksare neither completely randomnor completely regular Some-thing that we must emphasize is that most of the biologicaltechnological and social networks lie between these twoextremesThese systems are highly clustered as regular arrayshowever they have a characteristic path length like randomnetworks see [21] These types of networks are called small-world networks in analogy to the small-world phenomenon(popularly known as six degrees of separation)

In general terms these networks are regular arrange-ments made by connections or reconnections of pairs ofrandomly chosen nodes Even a very small number ofthese added connections commonly called shortcuts do notchange the local properties (very high clustering typical of aregular network) and cause those typical random networksvalues to be present in the average path length see [22]

Now we consider a complex network according to (1)-(2) composed of 119873 = 5 coupled fireflies periodic oscillators(7) as fundamental nodes in small-world coupled topologysee Figure 11 The corresponding state equations to thisdynamical network are arranged as follows the first firefly1198731is given by1199081sw (119896 + 1) = 1199081sw (119896) + 11990611sw (119896)

1199091sw (119896 + 1) = sin (1199081sw (119896) 119905 (119896))

11990611sw (119896) = 119888 (minus31199081sw (119896) +1199082sw (119896) +1199084sw (119896) + sdot sdot sdot

+1199085sw (119896))

(23)

the second firefly1198732 is given by1199082sw (119896 + 1) = 1199082sw (119896) + 11990621sw (119896)

1199092sw (119896 + 1) = sin (1199082sw (119896) 119905 (119896))

11990621sw (119896) = 119888 (minus21199082sw (119896) +1199081sw (119896) +1199083sw (119896))

(24)

the third firefly1198733 is given by1199083sw (119896 + 1) = 1199083sw (119896) + 11990631sw (119896)

1199093sw (119896 + 1) = sin (1199083sw (119896) 119905 (119896))

11990631sw (119896) = 119888 (minus21199083sw (119896) +1199082sw (119896) +1199084sw (119896))

(25)

the fourth firefly1198734 is given by1199084sw (119896 + 1) = 1199084sw (119896) + 11990641sw (119896)

1199094sw (119896 + 1) = sin (1199084sw (119896) 119905 (119896))

11990641sw (119896) = 119888 (minus31199084sw (119896) +1199081sw (119896) +1199083sw (119896) + sdot sdot sdot

+1199085sw (119896))

(26)

and the fifth firefly1198735 is given by1199085sw (119896 + 1) = 1199085sw (119896) + 11990651sw (119896)

1199095sw (119896 + 1) = sin (1199085sw (119896) 119905 (119896))

11990651sw (119896) = 119888 (minus21199085sw (119896) +1199084sw (119896) +1199081sw (119896))

(27)

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

10

minus1xixis

0 05 1 15 2 25 3 35

t(k)

xixis

(a)

6040200w

iwis

0 05 1 15 2 25 3 35

t(k)

wi

wis

(b)

20

minus2ximinusxis

0 05 1 15 2 25 3 35

t(k)

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

100

minus10wiminuswis

0 05 1 15 2 25 3 35

t(k)

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 15 Dynamics for two uncoupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and

errors (c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xi xis

(a)

0 05 1 15 2 25 3 35

t(k)

6040200w

iwis

wi wis

(b)

0 05 1 15 2 25 3 35

t(k)

0051

minus05ximinusxis

x1 minus x1sx2 minus x2sx3 minus x3s

x4 minus x4sx5 minus x5s

(c)

0 05 1 15 2 25 3 35

t(k)

10

0minus10w

iminuswis

w1 minus w1s

w2 minus w2s

w3 minus w3s

w4 minus w4s

w5 minus w5s

(d)

Figure 16 Dynamics for two coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) (b) 119908

119894(119896) 119908

119894119904(119896) and errors

(c) 119909119894(119896) minus 119909

119894119904(119896) and (d) 119908

119894(119896) minus 119908

119894119904(119896) where 119894 = 1 2 5

42 Star Coupled Network Now we consider a complexnetwork according to (1)-(2) composed of 119873 = 5 coupledfireflies periodic oscillators (7) as fundamental nodes instar coupled topology see Figure 9 The corresponding stateequations to this dynamical network are arranged as followsthe first firefly1198731 is given by

1199081119904 (119896 + 1) = 1199081119904 (119896) + 11990611119904 (119896)

1199091119904 (119896 + 1) = sin (1199081119904 (119896) 119905 (119896))

11990611119904 (119896) = 119888 (minus41199081119904 (119896) +1199082119904 (119896) + sdot sdot sdot +1199083119904 (119896)

+1199084119904 (119896) +1199085119904 (119896))

(16)

the second firefly1198732 is given by

1199082119904 (119896 + 1) = 1199082119904 (119896) + 11990621119904 (119896)

1199092119904 (119896 + 1) = sin (1199082119904 (119896) 119905 (119896))

11990621119904 (119896) = 119888 (minus1199082119904 (119896) +1199081119904 (119896))

(17)

the third firefly1198733 is given by

1199083119904 (119896 + 1) = 1199083119904 (119896) + 11990631119904 (119896)

1199093119904 (119896 + 1) = sin (1199083119904 (119896) 119905 (119896))

11990631119904 (119896) = 119888 (minus1199083119904 (119896) +1199081119904 (119896))

(18)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 17 Synchronization diagram for outer synchronization ofnearest-neighbor-star networks

the fourth firefly1198734 is given by

1199084119904 (119896 + 1) = 1199084119904 (119896) + 11990641119904 (119896)

1199094119904 (119896 + 1) = sin (1199084119904 (119896) 119905 (119896))

11990641119904 (119896) = 119888 (minus1199084119904 (119896) +1199081119904 (119896))

(19)

and the fifth firefly1198735 is given by

1199085119904 (119896 + 1) = 1199085119904 (119896) + 11990651119904 (119896)

1199095119904 (119896 + 1) = sin (1199085119904 (119896) 119905 (119896))

11990651119904 (119896) = 119888 (minus1199085119904 (119896) +1199081119904 (119896))

(20)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (21)

The corresponding coupling matrix A119904 for the networkshown in Figure 9 is given by

A119904 =

[

[

[

[

[

[

[

[

[

minus4 1 1 1 11 minus1 0 0 01 0 minus1 0 01 0 0 minus1 01 0 0 0 minus1

]

]

]

]

]

]

]

]

]

(22)

where subscript 119904 refers to the star network We carry out thenumerical simulations in the star network with 119888 = 001 andinitial conditions 1199081119904(0) = 212058725 1199082119904(0) = 212058755 1199083119904(0) =212058715 1199084119904(0) = 212058765 1199085119904(0) = 212058745 1199091119904(0) = 1199092119904(0) =1199093119904(0) = 1199094119904(0) = 1199095119904(0) = 0 119905(0) = 0 and 119896 = 600 Takinginto account the above conditions we obtain similar results asin the nearest-neighbor network and synchronization amongthe nodes was achieved Figure 10 shows the synchronizationdiagram where we can see the range 00021 lt 119888 lt 03995where the nodes in the network are synchronized

43 Small-World Coupled Network A network with complexcoupled topology can be represented by a random graph butour intuition tells us clearly that many real complex networksare neither completely randomnor completely regular Some-thing that we must emphasize is that most of the biologicaltechnological and social networks lie between these twoextremesThese systems are highly clustered as regular arrayshowever they have a characteristic path length like randomnetworks see [21] These types of networks are called small-world networks in analogy to the small-world phenomenon(popularly known as six degrees of separation)

In general terms these networks are regular arrange-ments made by connections or reconnections of pairs ofrandomly chosen nodes Even a very small number ofthese added connections commonly called shortcuts do notchange the local properties (very high clustering typical of aregular network) and cause those typical random networksvalues to be present in the average path length see [22]

Now we consider a complex network according to (1)-(2) composed of 119873 = 5 coupled fireflies periodic oscillators(7) as fundamental nodes in small-world coupled topologysee Figure 11 The corresponding state equations to thisdynamical network are arranged as follows the first firefly1198731is given by1199081sw (119896 + 1) = 1199081sw (119896) + 11990611sw (119896)

1199091sw (119896 + 1) = sin (1199081sw (119896) 119905 (119896))

11990611sw (119896) = 119888 (minus31199081sw (119896) +1199082sw (119896) +1199084sw (119896) + sdot sdot sdot

+1199085sw (119896))

(23)

the second firefly1198732 is given by1199082sw (119896 + 1) = 1199082sw (119896) + 11990621sw (119896)

1199092sw (119896 + 1) = sin (1199082sw (119896) 119905 (119896))

11990621sw (119896) = 119888 (minus21199082sw (119896) +1199081sw (119896) +1199083sw (119896))

(24)

the third firefly1198733 is given by1199083sw (119896 + 1) = 1199083sw (119896) + 11990631sw (119896)

1199093sw (119896 + 1) = sin (1199083sw (119896) 119905 (119896))

11990631sw (119896) = 119888 (minus21199083sw (119896) +1199082sw (119896) +1199084sw (119896))

(25)

the fourth firefly1198734 is given by1199084sw (119896 + 1) = 1199084sw (119896) + 11990641sw (119896)

1199094sw (119896 + 1) = sin (1199084sw (119896) 119905 (119896))

11990641sw (119896) = 119888 (minus31199084sw (119896) +1199081sw (119896) +1199083sw (119896) + sdot sdot sdot

+1199085sw (119896))

(26)

and the fifth firefly1198735 is given by1199085sw (119896 + 1) = 1199085sw (119896) + 11990651sw (119896)

1199095sw (119896 + 1) = sin (1199085sw (119896) 119905 (119896))

11990651sw (119896) = 119888 (minus21199085sw (119896) +1199084sw (119896) +1199081sw (119896))

(27)

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 17 Synchronization diagram for outer synchronization ofnearest-neighbor-star networks

the fourth firefly1198734 is given by

1199084119904 (119896 + 1) = 1199084119904 (119896) + 11990641119904 (119896)

1199094119904 (119896 + 1) = sin (1199084119904 (119896) 119905 (119896))

11990641119904 (119896) = 119888 (minus1199084119904 (119896) +1199081119904 (119896))

(19)

and the fifth firefly1198735 is given by

1199085119904 (119896 + 1) = 1199085119904 (119896) + 11990651119904 (119896)

1199095119904 (119896 + 1) = sin (1199085119904 (119896) 119905 (119896))

11990651119904 (119896) = 119888 (minus1199085119904 (119896) +1199081119904 (119896))

(20)

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (21)

The corresponding coupling matrix A119904 for the networkshown in Figure 9 is given by

A119904 =

[

[

[

[

[

[

[

[

[

minus4 1 1 1 11 minus1 0 0 01 0 minus1 0 01 0 0 minus1 01 0 0 0 minus1

]

]

]

]

]

]

]

]

]

(22)

where subscript 119904 refers to the star network We carry out thenumerical simulations in the star network with 119888 = 001 andinitial conditions 1199081119904(0) = 212058725 1199082119904(0) = 212058755 1199083119904(0) =212058715 1199084119904(0) = 212058765 1199085119904(0) = 212058745 1199091119904(0) = 1199092119904(0) =1199093119904(0) = 1199094119904(0) = 1199095119904(0) = 0 119905(0) = 0 and 119896 = 600 Takinginto account the above conditions we obtain similar results asin the nearest-neighbor network and synchronization amongthe nodes was achieved Figure 10 shows the synchronizationdiagram where we can see the range 00021 lt 119888 lt 03995where the nodes in the network are synchronized

43 Small-World Coupled Network A network with complexcoupled topology can be represented by a random graph butour intuition tells us clearly that many real complex networksare neither completely randomnor completely regular Some-thing that we must emphasize is that most of the biologicaltechnological and social networks lie between these twoextremesThese systems are highly clustered as regular arrayshowever they have a characteristic path length like randomnetworks see [21] These types of networks are called small-world networks in analogy to the small-world phenomenon(popularly known as six degrees of separation)

In general terms these networks are regular arrange-ments made by connections or reconnections of pairs ofrandomly chosen nodes Even a very small number ofthese added connections commonly called shortcuts do notchange the local properties (very high clustering typical of aregular network) and cause those typical random networksvalues to be present in the average path length see [22]

Now we consider a complex network according to (1)-(2) composed of 119873 = 5 coupled fireflies periodic oscillators(7) as fundamental nodes in small-world coupled topologysee Figure 11 The corresponding state equations to thisdynamical network are arranged as follows the first firefly1198731is given by1199081sw (119896 + 1) = 1199081sw (119896) + 11990611sw (119896)

1199091sw (119896 + 1) = sin (1199081sw (119896) 119905 (119896))

11990611sw (119896) = 119888 (minus31199081sw (119896) +1199082sw (119896) +1199084sw (119896) + sdot sdot sdot

+1199085sw (119896))

(23)

the second firefly1198732 is given by1199082sw (119896 + 1) = 1199082sw (119896) + 11990621sw (119896)

1199092sw (119896 + 1) = sin (1199082sw (119896) 119905 (119896))

11990621sw (119896) = 119888 (minus21199082sw (119896) +1199081sw (119896) +1199083sw (119896))

(24)

the third firefly1198733 is given by1199083sw (119896 + 1) = 1199083sw (119896) + 11990631sw (119896)

1199093sw (119896 + 1) = sin (1199083sw (119896) 119905 (119896))

11990631sw (119896) = 119888 (minus21199083sw (119896) +1199082sw (119896) +1199084sw (119896))

(25)

the fourth firefly1198734 is given by1199084sw (119896 + 1) = 1199084sw (119896) + 11990641sw (119896)

1199094sw (119896 + 1) = sin (1199084sw (119896) 119905 (119896))

11990641sw (119896) = 119888 (minus31199084sw (119896) +1199081sw (119896) +1199083sw (119896) + sdot sdot sdot

+1199085sw (119896))

(26)

and the fifth firefly1198735 is given by1199085sw (119896 + 1) = 1199085sw (119896) + 11990651sw (119896)

1199095sw (119896 + 1) = sin (1199085sw (119896) 119905 (119896))

11990651sw (119896) = 119888 (minus21199085sw (119896) +1199084sw (119896) +1199081sw (119896))

(27)

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

N1 N1

N2

N2

N3

N3

N4 N4

N5

N5

(a)

N1N2

N3

N4

N5

N1N2

N3

N4

N5

(b)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(c)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(d)

N1

N2N3

N4

N5

N1

N2N3

N4

N5

(e)

Figure 18 Cases for which outer synchronization was verified (a) small-world-star (b) star-star (c) small-world-nearest-neighbor (d)nearest-neighbor-nearest-neighbor and (e) small-world-small-world

N1 N1

N2

N2

N3

N3

N4 N4

N5

N1

N2N3

N4

N5

N5

Figure 19 Graph of outer synchronization for three coupled networks

with

119905 (119896 + 1) = 119905 (119896) + 120587

1000 (28)

The corresponding coupling matrix Asw for the networkshown in Figure 11 is given by

Asw =

[

[

[

[

[

[

[

[

[

minus3 1 0 1 11 minus2 1 0 00 1 minus2 1 01 0 1 minus3 11 0 0 1 minus2

]

]

]

]

]

]

]

]

]

(29)

where subscript sw refers to the small-world network

We carry out the numerical simulations in the small-world network with 119888 = 001 and initial conditions1199081sw(0) =212058723 1199082sw(0) = 212058753 1199083sw(0) = 212058713 1199084sw(0) = 2120587631199085sw(0) = 212058743 1199091sw(0) = 1199092sw(0) = 1199093sw(0) = 1199094sw(0) =1199095sw(0) = 0 119905(0) = 0 and 119896 = 600 Taking into account theabove conditions we obtain similar results as in the nearest-neighbor and star networks and synchronization among thenodes was achieved Figure 12 shows the synchronizationdiagram where we can see the range 00017 lt 119888 lt 04325where the nodes in the network are synchronized

With diffusive coupling a network topology plays animportant role in synchronization see [23 24] We cancorroborate this by analyzing and comparing the syn-chronization diagrams of the three topologies presentedso far

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3 35

t(k)

10

minus1xixis

xisw

xi xis xisw

(a)

0 05 1 15 2 25 3 35

t(k)

50

0wiwis

wisw

wi wis wisw

(b)

0 05 1 15 2 25 3 35

t(k)

050

minus05ximinusxis

ximinusxisw

x1 minus x1s x1 minus x1swx2 minus x2s x2 minus x2swx3 minus x3s x3 minus x3sw

x4 minus x4s x4 minus x4swx5 minus x5s x5 minus x5sw

(c)

0 05 1 15 2 25 3 35

t(k)

100

minus10

wiminuswis

wiminuswisw

w1 minus w1s w1 minus w1sww2 minus w2s w2 minus w2sww3 minus w3s w3 minus w3sw

w4 minus w4s w4 minus w4sww5 minus w5s w5 minus w5sw

(d)

Figure 20 Dynamics for three coupled networks with different topologies temporal dynamics of (a) 119909119894(119896) 119909

119894119904(119896) and 119909

119894sw(119896) and (b) 119908119894(119896)119908119894119904(119896) and 119908

119894sw(119896) and errors (c) 119909119894(119896) minus 119909

119894119904(119896) 119909

119894(119896) minus 119909

119894sw(119896) and (d) 119908119894(119896) minus 119908

119894119904(119896) 119908

119894(119896) minus 119908

119894sw(119896) where 119894 = 1 2 5

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

x1

xav

erag

e

Figure 21 Phase diagram xaverage versus 1199091 for three couplednetworks

44 Small-World Coupled Network inMass Now we use119873 =

50 nodes in small-world network with the only purpose toemulate the interaction of fireflies and its synchronization inmass

Starting from a nearest-neighbor coupled topology weadd 100 connections with the same probability to builda small-world network resulting in an average degree of119911 = 6 For obvious reasons we omit representations ofthe coupling matrix and graph on small-world network with119873 = 50 Figure 13 shows the synchronization with thesimple model proposed Initial conditions are 119909

119894sw(0) = 0and 119908

119894sw(0) = 2120587119891119894sw where 119891119894sw is randomly selected from

1 to 10Hz with 119894 = 1 2 50 119905(0) = 0 119888 = 001and 119896 = 600 Figure 13 shows the simulation results wheresynchronization is achieved for a small-world network with119873 = 50 nodes

5 Outer Synchronization

Outer synchronization occurs among coupled complex net-works which means that the corresponding nodes of the

1

08

06

04

02

0

minus02 0 02 04 06 08 1 12

c

Sync

hron

izat

ion=1

No

sync

hron

izat

ion=0

Figure 22 Synchronization diagram for outer synchronization ofnearest-neighbor-star-small-world networks

coupled networks will achieve synchronization [25ndash29] Theinteraction among communities (or networks) is somethingthat exists in our real world One important example is theinfectious disease that spreads among different communitiesFor example avian influenza spreads among domestic andwild birds afterward infecting human beings unexpectedlyAIDS mad cow disease bird flu and SARS were originallyspread between two communities (or networks) This meansthat the study of dynamics among coupled networks isnecessary and important [30] We use the complex networktheory that was shown in Section 2 to couple a pair of nodesbetween twonetworks andobtain synchronization among thenodes of the networks (outer synchronization)

51 Outer Synchronization of Two Networks Suppose twonetworks are the following one in nearest-neighbor coupledtopology and another in star coupled topology and thesenetworks are coupled as in Figure 14 First we show the casewhere the two networks are decoupled Figure 15 shows two

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

6VB1

C1

C3 C2

IC1

VR150k

100 kR1

R21k

R356

R433

D1

D2

D3

D4

D5

NE555

Q1 Q2 Q3 Q4

12

3

4

5

6

7 8

+

+

+ 100120583F

100 lx 100 lx 100 lx 100 lx 20120583F 20nF

Figure 23 Electronic circuit of a simple firefly

synchronization states belonging to each one of the networksthat is the nodes of each network are synchronized butthe two networks are unsynchronized Also we can observethat the angular frequencies of the two networks tend todifferent values so it is obvious that the two networks are notsynchronized If we couple the two networks that is if weconnect 1198731 of the nearest-neighbor network with 1198732 of thestar network and we use a coupling strength 119888 = 003 withinitial conditions as previously mentioned in Sections 41 and42 outer synchronization is obtained as shown in Figure 16where we clearly see that the dynamics of the states 119909

119894(119896)

and 119909119894119904(119896) as well as the states 119908

119894(119896) and 119908

119894119904(119896) tend to the

same value and errors among the states of the two networkswill be zero As expected the synchronization time for bothnetworks is greater than the synchronization time of a singlenetwork even with the increase of the coupling constant 119888 soin this case we simulate 119896 = 1000 iterations

Figure 17 shows the synchronization diagram where wecan see the range 00142 lt 119888 lt 03891 where the nodesamong the two networks are synchronized We perform thenumerical verification to the cases shown in Figure 18 with119888 = 003 and initial conditions as previously mentioned inSections 41 42 and 43 achieving synchronization in eachof the presented combinations

52 Outer Synchronization of the Three Different TopologiesTo complement the study of outer synchronization wepresent the outer synchronization among the networks thathave been presented in this paper

Consider three networks (Figures 3 9 and 11) We omitthe case where networks are decoupled Assume that thenetworks are coupled as in Figure 19 The correspondingresults for the coupling among the three networks (Figure 19)are shown in Figure 20 where 119888 = 005 119896 = 1000 and theinitial conditions are those previously used in Sections 4142 and 43 We also include a phase diagram in Figure 21where the abscissa axis is the average of the states x(119896) and

the ordinate axis is the state 1199091(119896) we use 119896 = 10000 in orderto eliminate the transient whereby outer synchronization canbe verified Figure 22 shows the synchronization diagramwhere we can see the range 00297 lt 119888 lt 03659 where thenodes among the three networks are synchronized

6 Electronic Implementation

Electronic firefly has been proposed in [31] to emulate thefirefly flashing process Nine circuits representing fireflieswere built and synchronized in [32] Here we reproduce suchcircuit and we coupled nine fireflies in different topologiesto realize the experimental fireflies synchronization A fire-fly can be constructed with electronic components as 555IC resistances capacitors LEDs IR and phototransistors(Figure 23) The nine electronic fireflies are in a 3 times 3 arrayconstructed in a 17 times 19 cm phenolic plate (Figure 24(a))Experimental results are shown in Figure 24 where synchro-nization of coupled fireflies in different topologies is achievedSynchronization was verified with the sensing pulses on adigital oscilloscope (Figure 25) [20] The synchronizationtime is instantaneous because coupling strength is consideredstrong based on [32]

7 Conclusions

In this paper network synchronization and outer synchro-nization problems with simple firefly discrete models havebeen investigated by using the theory of complex systemsThe simulation results show the effective synchronization andouter synchronization with the simple model proposed Allthe results presented help to understand how the collectivebehaviors work in nature and in living systems such asfireflies crickets chirping heart cell neurons and womenrsquosmenstrual cycles In addition experimental network syn-chronization with nine electronic fireflies was presented

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

(a) (b)

(c) (d)

Figure 24 Experimental fireflies synchronization (a) isolated nodes (b) line network synchronization with 119873 = 3 (c) synchronization inthree different networks with119873 = 4119873 = 3 and119873 = 2 and (d) network synchronization with119873 = 9

(a) (b)

Figure 25 Synchronization verification between two fireflies by means of LED pulsation comparison by using an oscilloscope (a) nosynchronization and (b) synchronization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the CONACYT Mexico underResearch Grant 166654

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

References

[1] J Buck ldquoSynchronous rhythmic flashing of fireflies IIrdquo Quar-terly Review of Biology vol 63 no 3 pp 265ndash289 1988

[2] HM Smith ldquoSynchronous flashing of firefliesrdquo Science vol 82no 2120 pp 151ndash152 1935

[3] M Bennett M F Schatz H Rockwood and K WiesenfeldldquoHuygensrsquos clocksrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2019 pp 563ndash579 2002

[4] A Goldbeter Biochemical Oscillations and Cellular RhythmsThe Molecular Bases of Periodic and Chaotic Behaviour Cam-bridge University Press 1996

[5] C Schafer M G Rosenblum H H Abel and J Kurths ldquoSyn-chronization in the human cardiorespiratory systemrdquo PhysicalReview E vol 60 no 1 pp 857ndash870 1999

[6] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[7] K M Cuomo A V Oppenheim and S H Strogatz ldquoSynchro-nization of Lorenz-based chaotic circuits with applications tocommunicationsrdquo IEEE Transactions on Circuits and Systems IIAnalog andDigital Signal Processing vol 40 no 10 pp 626ndash6331993

[8] S H Strogatz SYNC The Emerging Science of SpontaneousOrder Hyperion Press New York NY USA 2003

[9] A Pikovsky M Rosenblum and J Kurths Synchronization AUniversal Concept in Nonlinear Sciences Cambridge UniversityPress Cambridge UK 2001

[10] A T Winfree ldquoBiological rhythms and the behavior of popula-tions of coupled oscillatorsrdquo Journal of Theoretical Biology vol16 no 1 pp 15ndash42 1967

[11] C S PeskinMathematical Aspects of Heart Physiology CourantInstitute of Mathematical Sciences 1975

[12] J Grasman and M J W Jansen ldquoMutually synchronizedrelaxation oscillators as prototypes of oscillating systems inbiologyrdquo Journal of Mathematical Biology vol 7 no 2 pp 171ndash197 1979

[13] T Pavlidis Biological Oscillators Their Mathematical AnalysisAcademic Press 1973

[14] R E Mirollo and S H Strogatz ldquoSynchronization of pulse-coupled biological oscillatorsrdquo SIAM Journal on Applied Math-ematics vol 50 no 6 pp 1645ndash1662 1990

[15] U Ernst K Pawelzik and T Geisel ldquoDelay-inducedmultistablesynchronization of biological oscillatorsrdquoPhysical ReviewE vol57 no 2 pp 2150ndash2162 1998

[16] S H Strogatz R EMirollo and PMatthews ldquoCoupled nonlin-ear oscillators below the synchronization threshold relaxationby generalized Landau dampingrdquo Physical Review Letters vol68 no 18 pp 2730ndash2733 1992

[17] B Ermentrout ldquoAn adaptive model for synchrony in the fireflyPteroptyx malaccaerdquo Journal of Mathematical Biology vol 29no 6 pp 571ndash585 1991

[18] X F Wang and G Chen ldquoSynchronization in small-worlddynamical networksrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 12 no 1 pp 187ndash192 2002

[19] X F Wang ldquoComplex networks topology dynamics andsynchronizationrdquo International Journal of Bifurcation andChaosin Applied Sciences and Engineering vol 12 no 5 pp 885ndash9162002

[20] M A Murillo-Escobar R M Lopez-Gutierrez C Cruz-Hernandez and F Abundiz-Perez Sincronizacion de NueveLuciernagas Electronicas XX Engineering Architecture andDesign Days FIAD-UABC 2013

[21] M Newman A-L Barabusi and D JWatts EdsThe Structureand Dynamics of Networks Princeton Studies in ComplexityPrinceton University Press Princeton NJ USA 2006

[22] N Dorogovtsev and J F FMendes Evolution of Networks FromBiological Nets to the Internet and WWW Oxford UniversityPress Oxford UK 2003

[23] R C G Murguia R H B Fey and H Nijmeijer ldquoNetworksynchronization of time-delayed coupled nonlinear systemsusing predictor-based diffusive dynamic couplingsrdquo Chaos vol25 no 2 Article ID 023108 2015

[24] E SteurWMichiels H Huijberts andH Nijmeijer ldquoNetworksof diffusively time-delay coupled systems conditions for syn-chronization and its relation to the network topologyrdquo PhysicaD Nonlinear Phenomena vol 277 pp 22ndash39 2014

[25] Z Li and X Xue ldquoOuter synchronization of coupled networksusing arbitrary coupling strengthrdquo Chaos vol 20 no 2 ArticleID 023106 7 pages 2010

[26] J Feng S Sheng Z Tang and Y Zhao ldquoOuter synchroniza-tion of complex networks with nondelayed and time-varyingdelayed couplings via pinning control or impulsive controlrdquoAbstract and Applied Analysis vol 2015 Article ID 414596 11pages 2015

[27] X Fang Q Yang andW Yan ldquoOuter synchronization betweencomplex networks with nonlinear time-delay characteristicsand time-varying topological structuresrdquo Mathematical Prob-lems in Engineering vol 2014 Article ID 437673 10 pages 2014

[28] S Zheng ldquoAdaptive impulsive observer for outer synchro-nization of delayed complex dynamical networks with outputcouplingrdquo Journal of Applied Mathematics vol 2014 Article ID450193 11 pages 2014

[29] D Ning X Wu J-a Lu and H Feng ldquoGeneralized outersynchronization between complex networks with unknownparametersrdquoAbstract and Applied Analysis vol 2013 Article ID802859 9 pages 2013

[30] C Li C XuW Sun J Xu and J Kurths ldquoOuter synchronizationof coupled discrete-time networksrdquo Chaos vol 19 no 1 ArticleID 013106 2009

[31] WGarver andFMoss ldquoElectronic firefliesrdquo ScientificAmericanvol 269 no 6 pp 128ndash130 1993

[32] T-H Chuang C-Y Chang J Byerly and C Slattery Synchro-nized Electronic Fireflies 2014 httpwww2eceohio-stateedusimpassinofireflyfolderfireflyhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of