dynamics, chaos control, and synchronization in a fractional-order samardzija...

Research ArticleDynamics, Chaos Control, and Synchronization in aFractional-Order Samardzija-Greller Population System withOrder Lying in (0, 2)

A. Al-khedhairi,1 S. S. Askar,1,2 A. E. Matouk ,3 A. Elsadany,4 and M. Ghazel5

1Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455,Riyadh 11451, Saudi Arabia2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt3Department of Basic Engineering Sciences, College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia4Mathematics Department, College of Sciences and Humanities Studies Al-Kharj, Prince Sattam Bin Abdulaziz University,Al-Kharj, Saudi Arabia5Mathematics Department, Faculty of Science, Hail University, Hail 2440, Saudi Arabia

Correspondence should be addressed to A. E. Matouk; [email protected]

Received 28 January 2018; Revised 3 July 2018; Accepted 16 July 2018; Published 10 September 2018

Academic Editor: Matilde Santos

Copyright © 2018 A. Al-khedhairi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper demonstrates dynamics, chaos control, and synchronization in Samardzija-Greller population model with fractionalorder between zero and two. The fractional-order case is shown to exhibit rich variety of nonlinear dynamics. Lyapunovexponents are calculated to confirm the existence of wide range of chaotic dynamics in this system. Chaos control in this modelis achieved via a novel linear control technique with the fractional order lying in (1, 2). Moreover, a linear feedback controlmethod is used to control the fractional-order model to its steady states when 0 < α < 2. In addition, the obtained resultsillustrate the role of fractional parameter on controlling chaos in this model. Furthermore, nonlinear feedback synchronizationscheme is also employed to illustrate that the fractional parameter has a stabilizing role on the synchronization process in thissystem. The analytical results are confirmed by numerical simulations.

1. Introduction

Dynamic analysis of engineering and biological models hasbecome an important issue for research [1–10]. One of themost fascinating dynamical phenomena is the existence ofchaotic attractors. Due to the importance of chaos, it has beeninvestigated in various academic disciplines [11–15]. Thesensitivity to initial conditions which characterizes the exis-tence of chaotic attractors was first noticed by Poincaré [16].

According to their potential applications in a wide varietyof settings, fractional-order differential equations (FODEs)have received increasing attention in engineering [17–23],physics [24], mathematical biology [25-26], and encryptionalgorithms [27]. Moreover, FODEs play an important rolein the description of memory which is essential in mostbiological models.

Chaos synchronization and control in dynamical systemsare essential applications of chaos theory. They have becomefocal topics for research since the elegant work of Ott et al. inchaos control [28] and the pioneering work of Pecora andCarroll in chaos synchronization [29]. Chaos control issometimes needed to refine the behavior of a chaotic modeland to remove unexpected performance of power electronics.Synchronization of chaos has also useful applications tobiological, chemical, physical systems and secure communi-cations. Furthermore, synchronization and control in chaoticfractional-order dynamical systems have been investigatedby authors [30–32].

The integer-order Samardzija-Greller population modelis a system of ODEs that generalizes the Lotka-Volterra equa-tions and expresses the behaviors of two species predator-prey population dynamical system. This model was proposed

HindawiComplexityVolume 2018, Article ID 6719341, 14 pageshttps://doi.org/10.1155/2018/6719341

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by Samardzija and Greller in 1988 [33]. They had proved theexistence of complex oscillatory behavior in this model. In1999, chaos synchronization had firstly been investigated inthis model by Costello [34]. Oancea et al. utilized this systemto achieve the pest control in agricultural systems [35]. In2018, Elsadany et al. proved the existence of generalizedHopf (Bautin) bifurcation in this model [36]. Recently,some works investigating synchronization in the fractional-order Samardzija-Greller model with order less than onehave appeared [37-38]. Up to the present day, dynamics,chaos control, and synchronization have not been investi-gated in the fractional-order Samardzija-Greller system withorder lying in (0, 2).

So, our objective in this paper is to investigate the richdynamics and achieve chaos synchronization and control inthe fractional-order Samardzija-Greller model with orderlying in (0, 2). Moreover, applying the fractional Routh-Hurwitz criteria [23, 39] enables us to illustrate the role ofthe fractional parameter on synchronizing and controllingchaos in this model. Motivated by the previous discussion,numerical verifications are performed to show the existenceof wide range of chaotic dynamics in this model using theaids of phase portraits and Lyapunov exponents.

2. Mathematical Preliminaries

Here, Caputo definition [17] is adopted as

Dα f t = Jl−α f l t , 1

where α ∈ℜ+, f l denotes the ordinary lth derivative of f t ,l is an integer that satisfies l − 1 < α ≤ l, and the operator Jβ isdefined by

Jβg t = 1Γ βt

0t − τ β−1g τ dτ, β > 0, 2

where Γ is defined as

Γ ς =∞

0tς−1e−tdt 3

Therefore, fractional modeling provides more accuracyin both theoretical and experimental results of the ecologicalmodel which are naturally related with long range memorybehavior which is very important in modeling ecologicalsystems. Thus, increasing the range of the fractional parame-ter α from the interval (0, 1) that is commonly used in mostliteratures to the interval (0, 2) provides greater degrees offreedom in modeling the population system. In addition,increasing the interval of fractional parameter increases thecomplexity in the system since the fractional parameter α isin a close relationship with fractals which are abundant inecosystems. Furthermore, as α is increased, more adequatedescription of the whole time domain of the process isachieved and the system may show rich variety of complexbehaviors such as sensitive dependence on initial conditions.

Consider the following fractional-order system:

DαX = JX, 4

which represents a linearized form of the nonlinear system:

DαX =G X , 5

where 0 < α < 2, X ∈ℜn, G ∈ C Ψ,ℜn . Moreover, the follow-ing inequalities [40]

arg λi >απ

2 , i = 1,… , n, 6

determine the local stability of a steady state X ∈ℜn of thefractional-order system (4), where λi is any eigenvalue ofthe matrix J. The stability region for α ∈ 0, 1 and α ∈ 1, 2is shown in Figures 1(a) and 1(b), respectively. Moreover,stability conditions and their applications to nonlinear sys-tems of FODEs were investigated in [23, 41].

3. A Three-Dimensional Samardzija-Greller Model

A three-dimensional Samardzija-Greller model [33] isdescribed as

dxdt

= 1 − y x + c − az x2,

dydt

= −1 + x y,

dzdt

= −b + ax2 z,

7

where the prey of the population is denoted by x and thepredators of the population are denoted by y and z. Thepredators y, z do not interact directly with one anotherbut compete for prey x, and a, b, c are nonnegative param-eters used to discuss the bifurcation phenomena in themodel as stated by Samardzija and Greller in 1988 [33].Furthermore, the positive value of the parameter a indi-cates that two different predators y and z consume theprey x. However, if parameter a is vanished, a classicalversion of Lotka-Volterra model is obtained. By replacingfirst-order derivatives in system (7) with fractional deriva-tives of order 0 < α < 2, we obtain

Dαx = 1 − y x + c − az x2,Dαy = −1 + x y,Dαz = −b + ax2 z

8

So, a greater degree of accuracy can be obtained in ourpopulation model by using the property of evolution (8)due to the existence of fractional derivatives which isessential to the proposed Samardzija-Greller model

2 Complexity

because of its ability to provide a realistic description ofthe population model which involves processes with mem-ory and hereditary properties. Furthermore, fractionalderivatives provide greater degrees of freedom in modelingpredator-prey ecosystems and closely related to fractalswhich are abundant in population models as well as itsability to describe the whole time domain for the process.

To obtain the steady states of system (8), we set Dαx = 0,Dαy = 0,Dαz = 0, then the system has three nonnegativeequilibria given as E0 = 0,0,0 , E1 = 1, 1 + c, 0 , and E2 =m, 0, 1 +mc /ma , where m = b/a.

4. Stability of System (8) with α Lying in (0, 2)

Assume that system (8) is written in the following form:

Dαx1 t = f1 x1, x2, x3 ,Dαx2 t = f2 x1, x2, x3 ,Dαx3 t = f3 x1, x2, x3 ,

9

α ∈ 0, 2 . The characteristic polynomial of the steady stateX∗ of system (9) is given as

P λ = λ3 + r1λ2 + r2λ + r3 = 0 10

The discriminant of P λ is given by

D P = 18r1r2r3 + r21r22 − 4r3r31 − 4r32 − 27r23 11

If all roots of (10) satisfy the inequalities (6), then thesteady state of the linearized form of system (9) is locallyasymptotically stable (LAS). To discuss the local stability ofX∗, we prove the following theorem:

Theorem 1. The steady state X∗ of system (9) is LAS if argλi J X∗ > απ/2, i = 1,2,3, α ∈ 0, 2 , where J is the Jaco-bian matrix computed at the steady state X∗.

Proof 1. Let si be a very small perturbation defined as si =xi − x∗i , where i = 1,2,3. Therefore, (9), with derivatives inthe Caputo sense, can be written as

Dαsi t = f i si + x∗i , 12

with initial values si 0 = xi 0 − x∗i . A linearization of theabove equation, based on the Taylor expansion, can takethe form

Dαs = Js, 13

where α ∈ 0, 2 , J is computed at the steady state X∗ = x∗1 ,x∗2 , x∗3 , and s = s1 s2 s3

T . Hence,

Dαs = PQP−1 s,

Dα P−1s =Q P−1s ,14

where Q = diag λ1, λ2, λ3 , λi is an eigenvalue of J, and P isthe corresponding eigenvector. Suppose that γ = P−1s =γ1 γ2 γ3

T , then it follows that

Dαγ =Qγ,Dαγi = λiγi

15

The above system can easily be solved by the aid ofMittag-Leffler functions as follows

γi t = 〠∞

k=0

tαkλkiΓ 1 + αk γi 0 , i = 1,2,3 16

If the eigenvalue λi satisfies the condition argλi J X∗ > απ/2, then the perturbations si t are decreas-ing which implies that X∗ is LAS.

Moreover, the fractional Routh-Hurwitz (FRH) stabilitycriterion is provided by the following lemma:

�훼�휋/2 Re (�휆)

Im (�휆)

−�훼�휋/2

Unstable

Unstable

Stable

Stable

Stable

(a)

Stable

Unstable

Unstable Unstable

Unstable

StableRe (�휆)

�훼�휋/2

−�훼�휋/2

Im (�휆)

(b)

Figure 1: Stability and instability regions of system (4): (a) the case 0 < α < 1; (b) the case 1 ≤ α < 2.

3Complexity

Lemma 1 (see [39]).

(i) The steady state X∗ is LAS iff

r1 > 0,r3 > 0,

r1r2 > r3,17

provided that the discriminant of (10) is positive.

(ii) If the discriminant of (10) is negative, and r1 ≥ 0,r2 ≥ 0, r3 > 0 , then the steady state X∗ is LAS whenthe fractional parameter α is less than 2/3, whileif fractional parameter α is greater than 2/3, andr1 < 0, r2 < 0, then X∗ is not LAS.

(iii) If the discriminant of (10) is negative, and r1 > 0,r2 > 0, r1r2 = r3, then X∗is LAS when 0 < α < 1.

(iv) Point X∗ is LAS only if r3 > 0.

Hence, the following lemmas are provided.

Lemma 2 (see [23, 39]). If the discriminant of (10) is negativeand r1r2 = r3, r1 > 0, r2 > 0, then the steady state X∗ is LASjust when 0 < α < 1.

Lemma 3 (see [42]). Assume that system (5) is written in theform

DαX t =CX t + h X t , 18

where α ∈ 0, 2 , C ∈ℜn×n is a constant matrix, and h X t isa nonlinear vector function such that

limX t →0

h X tX t

= 0, 19

where represents the l2-norm, then the zero steady stateof system (5) is LAS provided that arg λi C > απ/2,i = 1,… , n.

4.1. Stability Conditions for the Steady State E0. The charac-teristic equation of the steady state E0 is described as

P λ = λ3 + bλ2 − λ − b = 0 20

It is easy to check that r3 = −b < 0, so by applying thestability condition (iv), we conclude that the steady state E0of system (8) is unstable.


P λ = λ3 + b − a − c λ2 + 1 + ac − bc + c λ+ c + 1 b − a = 0

21

So by utilizing the FRH conditions (i)–(iv), we obtain thefollowing results:

(1) If the discriminant of (21) is negative; the steadystate E1 is LAS provided that a + c < b, b ≤ a + 1 +1/c, α < 2/3.

However, if a + c > b, b > a + 1 + 1/c, α > 2/3, the steady stateE1 is unstable.

(2) The steady state E1 of system (8) is LAS only if b > a.

Remark 1. The stability conditions (i) and (iii) are not satis-fied for the steady state E1.


P λ = λ3 + 2 −m λ2 + 2b + 1 + 2mbc −m λ+ 2b mc + 1 1 −m = 0,

22

where m = b/a. So by utilizing the FRH conditions (i)–(iv),we obtain the following results:

(1) When the discriminant of (22) is positive, the steadystate E2 of system (8) is LAS iff

b < a,

c > 3ma − 2a − b − 2ab2mab23

(2) When the discriminant of (22) is negative, the steadystate E2 is LAS provided that b < a, c ≥m − 2b − 1/2mb, α < 2/3. However, the steady state E2 is unstableif b > 4a, c < m − 2b − 1 /2mb, α > 2/3. Moreover, ifthe discriminant of (22) is negative, b < 3a, then E2is LAS for all 0 < α < 1.

(3) The steady state E2 of system (8) is LAS only if b < a.

5. Hopf Bifurcation in Samardzija-GrellerModel (8)

Although exact periodic solutions do not exist in auton-omous fractional systems [43], some asymptotically peri-odic signals converge to limit cycles have been observed bynumerical simulations in many fractional systems. Theselimit cycles attract the nearby solutions of such systems.Therefore, Hopf bifurcation (HB) in autonomous fractionalsystems is expected to occur around an equilibrium pointof such systems if it has a pair of complex conjugate eigen-values and at least one negative eigenvalue. However, deter-mining the precise bifurcation type is difficult.

In Samardzija-Greller system (8), the occurrence of HB isexpected around E1 = 1, 1 + c, 0 at the critical parameter

c∗ = 2 + 2 2 + tan2 απ/2

1 + tan2 απ/2 ,24

4 Complexity

at which the transversality condition holds and 2 1 − 2< c < 2 1 + 2 , a < b. For α = 0 98, a = 0 8, b = 3, and c =0 07, system (8) converges to a limit cycle as shown inFigure 2.

6. Chaos in Samardzija-Greller Model (8)

Based on the algorithm given in [44] and using the parameterset a, b, c = 8/10, 3, 3 , the maximum values of Lyapunovexponents (MLEs) of system (8) are computed. The calcula-tions of the MLEs give the values 0.21315, 0.038468, 0.0089,and 0.1562 when α = 1 1, α = 1 05, α = 1 00, and α = 0 99,respectively. In Figure 3, it is shown that the chaoticattractor of system (8) related to the fractional-order caseis more complicated than the attractor related to theinteger-order case.

Another parameter set a, b, c = 3, 4, 10 is used toexplore a variety of complex dynamics and new chaoticregions in the Samardzija-Greller model (7) and its corre-sponding fractional-order form. The results are depicted inFigure 4 which shows that the complex dynamics of thesystem exist in a wide range of the fractional parameter α.These foundations are confirmed by the calculation of maxi-mal Lyapunov exponents which are depicted in Figure 5.

7. Chaos Control in System (8)

In the following, we will control chaos in system (8) when1 < α < 2 using linear control technique and also we willdiscuss the role of fractional parameter α on controllingchaos in Samardzija-Greller model using a linear feedbackcontrol technique.

7.1. Chaos Control of Samardzija-Greller System (8) with αLying in (1, 2) via Linear Control. Assume that the controlledSamardzija-Greller system is described by

Dαx = 1 − y x + c − az x2 − u1,Dαy = y x − 1 − u2,Dαz = −b + ax2 z − u3

25

Thus, system (25) is rewritten as

DαX t =CX t + h X t −U , 26

where X t = x y z T ,

C =1 0 00 −1 00 0 −b

27

is a constant parameter matrix, and the linear controller U iswritten as

U =u1

u2

u3

=MKX, 28

1.41.2

10.8 0.7 0.8

0.9y x11.1 1.2

1.3 1.4

0

0.005

0.01

0.015

0.02

z

(a)

10080 9070603010 40 50200t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x, y, z

x

y

z

(b)

Figure 2: The trajectory of system (8) with the parameter set a, b, c = 0 8,3,0 07 and fractional parameter α = 0 98 is attracted by a limitcycle: (a) view in xyz-space; (b) the solution versus time.

5Complexity

where U is to be designed later, and h X t is also defined as

h X t =−xy + cx2 − ax2z

xy

ax2z

29

Obviously,

limX t →0

h X tX t

= limX t →0

−xy + cx2 − ax2z 2 + x2y2 + a2x4z2

x2 + y2 + z2

≤ limX t →0

x2 −y + cx − axz 2 + y2 + a2x2z2

x2

= limX t →0

−y + cx − axz 2 + y2 + a2x2z2 = 0

30

So, according to Lemma 3, the zero steady state x − x∗,y − y∗, z − z∗ of the controlled system (26) is LAS if the

linear control matrix is chosen such that arg λi C −U> απ/2, i = 1,2,3.

The controlled Samardzija-Greller model (25) is numer-ically integrated as a = 3, b = 4,c = 10, and α = 1 1 and linearcontrol functions

u1 = k1 x − x∗ ,u2 = k2 y − y∗ ,u3 = k3 z − z∗ + ε x − x∗ ,

31

where k1, k2, k3 ≥ 0 and ε is a real constant. Thus, theselection k1 > 1, k2 > 0, k3 > 0 ensures that the conditionsarg λi C −U > απ/2, i = 1,2,3, hold. Figures 6(a)–6(c)show that system (26) is controlled to the steady states E0,E1, E2 as using (k1 = 100, k2 = 150, k3 = 100, and ε = 100),(k1 = 100, k2 = 150, k3 = 150, and ε = 0 3), (k1 = 100, k2 =100, k3 = 150, and ε = 0 3), respectively.

7.2. Chaos Control of Samardzija-Greller System (8) via theFRH Criterion. The FRH criterion is employed to controlchaos in system (8) using linear feedback control techniquewhich is more easy of implementation, cheap in cost, and

25

20

15

10

5

01.5

1

0.50 −2 0

24

x6 8

10 12

y

z

(a)

6

45

21

3

0y

z

x0

24

68

02468101214

(b)

05

43

21y

z

x0 0 510

15 2025

30

510152025303540

(c)

Figure 3: Phase diagrams of Samardzija-Greller model (8) with a = 8/10, b = c = 3, and α equals: (a) 1.1, (b) 1.00, and (c) 0.99.

6 Complexity

more appropriate of being designed in real-world applica-tions.Moreover, wewill illustrate the role of fractional param-eter α on controlling chaos in system (8). Consider thefollowing system:

DαX = G X , 32

where α ∈ 0, 2 and X = x, y, z . System (32) has the con-trolled form

DαX = G X − K X − X∗ , 33

where X∗ = x∗, y∗, z∗ is a steady state of (32) and K = diagk1, k2, k3 , k1, k2, k3 ≥ 0 are the feedback control gains

10

8

6

4

y

0 0.5 1 x1.5 2.52

3 3.54.54

0

2

4

6

8

10

z

(a)

0.3

0.25

0.2

0.15

0.1

y

−1 0 12 3 4

5 6 78 9

x

0

2

4

6

8

10

12

14

16

z

(b)

02

5

10

15

20

z

1.5

1

0.5

y

00 2 4 x

6 8 1012 14

(c)

25

20

15

z

10

5

06

4

2

0

y

0 2 46

x8 10 12

14 16 18

(d)

15

10

5

z

06

5

4

3

2

1

y

0 1 23

x4 5 6 7

8 9 10

(e)

8

7

6

5

4

3

2

1

0

z

5

4.5

4

3.5

3

y

0 0.5 11.5 2 2.5

3 3.5 44.5

x

(f)

4.5

4

3.5

3

2.5

z

2

1.5

1

0.57

6

5

4

y

0.5 11.5

x

2 2.53 3.5

(g)

z

6

5

4

3

2

1

010

8

6

4

y

0 12

x3 4

5 6

(h)

6

5

4

3

2

1

0

z

10

8

6

4

y

0 12

x3 4

5 6

(i)

3

2.5

2

1.5z

1

0.510

8

6y

40.7 0.8 0.9

1 1.1 1.21.3 1.4 1.5

1.6

x

(j)

Figure 4: Phase diagrams of Samardzija-Greller model (8) with α = 3, b = 4, c = 10, and α equals: (a) 1.1, (b) 1.05, (c) 1.00, (d) 0.98, (e) 0.95, (f)0.90, (g) 0.80, (h) 0.70, (i) 0.60, and (j) 0.45.

7Complexity

(FCGs) that are selected according to the FRH criterion insuch a way that

limt→∞

X − X∗ = 0 34

So, controlled Samardzija-Greller system (8) is given as

Dαx = 1 − y x + c − az x2 − k1 x − x∗ ,Dαy = −1 + x y − k2 y − y∗ ,Dαz = −b + ax2 − k3 z + k3z∗,

35

where 0 < α < 2 and X∗ = x∗, y∗, z∗ is a steady state of sys-tem (8). Now, we are going to control chaotic system (8) tothe steady state E1. Hence, system (35) becomes

Dαx = 1 − y x + c − az x2 − k1 x − 1 ,Dαy = −1 + x y − k2 y − 1 − c ,Dαz = −b + ax2 − k3 z

36

Then, system (36) has the following characteristic equa-tion that is evaluated at the steady state E1:

P λ = λ3 + r1λ2 + r2λ + r3 = 0, 37

where

r1 = k1 + b − a + c ,r2 = 1 + c + ac − ak1 + bk1 − bc,r3 = 1 + c b − a

38

System (36) is numerically integrated with a = 3, b = 4,c = 10, α = 0 95, and k1 = 8 5, k2 = 1 5, k3 = 50. Consequently,the fractional Routh-Hurwitz condition (iii) holds, that is, thediscriminant of (37) is negative, r1, r2 are positive and r3 =r1r2. So according to Lemma 2, system (36) is stabilized to

the steady state E1 = 1,11,0 with the fractional parameterα only in the interval 0, 1 . Figure 7(a) shows that the statesof Samardzija-Greller model (36) approach the steady stateE1 = 1,11,0 . However, Figures 7(b) and 7(c) show that thestates of system (36) are not stabilized to the steady stateE1 = 1,11,0 when α = 1 and α = 1 1, respectively. Theseresults illustrate the role of the parameter α on suppressingchaos in Samardzija-Greller model (8).

Obviously, E0 satisfies the fractional Routh-Hurwitz con-dition (i) as using the above selection of parameters andFCGs k1 > 1, k2 = 1 5, k3 = 50 and k1 = 8 5, k2 > 0, k3 = 50.So, system (35) is controlled to E0 for α = 0 95, k1 = 8 5,k2 = 1 5, k3 = 50 and for α = 1 1, k1 = 8 5, k2 = 250, k3 = 50.The results are depicted in Figure 8.

Also, E2 = 1 154700539,0,3 622008471 satisfies thefractional Routh-Hurwitz condition (i) as using the aboveselection of parameters and FCGs k1 > 0, k2 = 1 5, k3 = 50and k1 = 8 5, k2 > 0 1547005382, k3 = 50. Hence, system (35)is controlled to E2 = 1 154700539,0,3 622008471 for α =0 95, k1 = 8 5, k2 = 1 5, k3 = 50 and for α = 1 1, k1 = 8 5, k2 =250, k3 = 50. The results are illustrated in Figure 9.

8. Chaos Synchronization of Samardzija-GrellerSystem (8) via Nonlinear Feedback Control

The drive system is introduced as

Dαx1 = 1 − y1 x1 + c − az1 x21,Dαy1 = −1 + x1 y1,Dαz1 = −b + ax21 z1,

39

and the response system is presented as

Dαx2 = 1 − y2 x2 + c − az2 x22 + v1,Dαy2 = −1 + x2 y2 + v2,Dαz2 = −b + ax22 z2 + v3,

40

where α ∈ 0, 2 and v1, v2, v3 are the nonlinear feedback con-trollers. Then, we suppose that

e1 = x2 − x1,e2 = y2 − y1,e3 = z2 − z1

41

By subtracting (39) from (40) and using (41), we get

Dαe1 = e1 − x2e2 + y1e1 + c x2 + x1 e1− a x22e3 + z1 x2 + x1 e1 + v1 e1, e2, e3 ,

Dαe2 = −e2 + x2e2 + y1e1 + v2 e1, e2, e3 ,

Dαe3 = −be3 + a x22e3 + z1 x2 + x1 e1 + v3 e1, e2, e342

Theorem 2. The drive chaotic system (39) and the response

1.10

0.05

0.1

0.15

0.2

0.25

Max

imal

lyap

unov

expo

nent

s

0.55 0.65 0.75 0.85 0.95 1.050.45Alpha

Figure 5: The MLEs of system (8) for different values of theparameter α and a, b, c = 3, 4, 10 .

8 Complexity

chaotic system (40) are synchronized for 0 < α < 2, if the con-trol functions are chosen as

v1 e1, e2, e3 = u − cv + aw − k1e1,v2 e1, e2, e3 = −u − k2e2,v3 e1, e2, e3 = −aw − k3e3,

43

where

u = x2e2 + y1e1,v = x2e1 + x1e1,w = x22e3 + x2z1e1 + x1z1e1,

44

and the FCGs must satisfy

k1 > 1,k2 > 0,k3 > 0

45

Proof 2. The Jacobian matrix of system (42) with the control-lers (43) evaluated at the origin steady state is described by

A 0,0,0 =1 − k1 0 00 −1 − k2 00 0 −b − k3

46

So, if we select k1 > 1, k2 > 0, k3 > 0, then all the eigenvaluesof the Jacobian matrix (46) have negative signs. Thus, accord-ing to condition (6), the origin steady state of system (42) isLAS for 0 < α < 2. Consequently, the proof is completed.

The fractional-order systems (39) and (40) are numer-ically integrated using the system’s parameters a = 8/10,b = 3, and c = 3, the controllers (43) with k1 = 20, k2 = 20,k3 = 20, and the fractional parameters α = 0 99 and α = 1 1,respectively. The results are depicted in Figure 10.

8.1. The Role of Fractional Parameter on Synchronizing Chaosin System (8) via Nonlinear Feedback Control Method. To

−0.1

0

0.1

0.2

0.3

0.4

0.5x, y, z

10 20 30 40 50 60 70 80 90 1000t

(a)

−2

0

2

4

6

8

10

12

x, y, z

0 20 30 40 50 60 70 80 90 10010t

(b)

0

0.5

1

1.5

2

2.5

3

3.5

4

x, y, z

10 20 30 40 50 60 70 80 90 1000t

(c)

Figure 6: The states of the controlled Samardzija-Greller model (25) converge to the steady state: (a) E0 = 0, 0, 0 , (b) E1 = 1, 11, 0 , and (c)E2 = 1 154700539, 0, 3 622008471 , when a = 3, b = 4, c = 10, and α = 1 1 and using the linear controllers (31).

9Complexity

explain the role of fractional parameter α on the synchroniza-tion process of this model, we present the following theorem:

Theorem 3. The trajectories of the drive chaotic Samardzija-Greller model (39) asymptotically approach the trajectoriesof the response chaotic Samardzija-Greller model (40) justwhen the fractional parameter α is less than one, if the controlfunctions are chosen as

v1 e1, e2, e3 = b − 1 − k1 e1 − e2 − ae3 + u − cv + aw,v2 e1, e2, e3 = 1 + c e1 + 1 − k2 e2 − u,v3 e1, e2, e3 = −aw − k3e3,

47where a = 3, b = 4, c = 10, k1 = 3 9, k2 = 0 1, and k3 = 10.

Proof 3. The Jacobian matrix of system (42) with the control-lers (47) evaluated at the origin steady state is described by

A 0,0,0 =b − k1 −1 −a1 + c −k2 00 0 −b − k3

48

and has the following characteristic equation:

P λ = λ3 + a1λ2 + a2λ + a3 = 0, a1 = k1 + k2 + k3,

a2 = −b2 + b k1 − k3 + c + 1 + k1k2 + k2k3 + k1k3,a3 = b + k3 c + 1 − b2k2 + bk2 k1 − k3 + k1k2k3

49

0

5

10

15

20

25

30

35x, y, z

5 10 15 20 25 30 35 400t

xyz

(a)

−5

0

5

10

15

20

x, y, z

10 20 30 40 50 60 800 70t

xyz

(b)

−5

0

5

10

15

20

x, y, z

80100 20 30 60 705040t

xyz

(c)

Figure 7: The states of the controlled Samardzija-Greller model (36) (a) converge to the steady state E1 = 1, 11, 0 when α = 0 95, (b) are notstabilized to the steady state E1 = 1, 11, 0 when α = 1, (c) are not stabilized to the steady state E1 = 1, 11, 0 when α = 1 1, using theparameter values a = 3, b = 4, and c = 10 and FCGs k1 = 8 5, k2 = 1 5, and k3 = 50.

10 Complexity

Using the parameters a = 3, b = 4, and c = 10 and the con-trollers (47) along with the selection k1 = 3 9, k2 = 0 1, k3 = 10, the characteristic equation (49) satisfies the fractionalRouth-Hurwitz condition (iii), since its discriminant is nega-tive, a1 ∈ℜ+, a2 ∈ℜ+, and a3 = a1a2. So according to Lemma2, the origin steady state of system (42) is LAS only forall α ∈ 0, 1 . Consequently, the states of Samardzija-Greller model (39) asymptotically approach the states ofSamardzija-Greller model (40) as the controllers (47) areemployed. Since the characteristic equation (49) has a pair

of purely imaginary eigenvalues, the condition of asymptoticstability of the origin steady state 0,0,0 of (42) is not satis-fied as 1 ≤ α < 2. Thus, the theorem is now proved.

Therefore, as the parameter α is decreased, the nonlinearfeedback control technique has a stabilizing effect on thesynchronization process in the chaotic Samardzija-Grellersystems. These results are illustrated in Figure 11.

On the other hand, the origin steady state 0,0,0 of (42)satisfies the fractional Routh-Hurwitz condition (i) as usinga = 3, b = 4, and c = 10 and FCGs k1 > 3 9, k2 = 0 1, k3 = 10.

xyz

0

2

4

6

8

10

12

14x, y, z

5 10 15 20 25 30 35 400t

(a)

xyz

−5

0

5

10

15

20

25

30

x, y, z

0 20 30 40 50 60 70 80 9010 100t

(b)

Figure 8: The trajectories of Samardzija-Greller model (35) with a = 3, b = 4, and c = 10 are controlled to E0 = 0, 0, 0 as (a) α = 0 95, k1 = 8 5,k2 = 1 5, and k3 = 50; (b) α = 1 1, k1 = 8 5, k2 = 250, and k3 = 50.

0

1

2

3

4

5

6

7

x, y, z

5 10 15 20 25 30 35 400t

xyz

(a)

0

1

2

3

4

5

6

7

8

x, y, z

5 10 15 20 25 30 35 400t

xyz

(b)

Figure 9: The trajectories of Samardzija-Greller model (35) with a = 3, b = 4, and c = 10 are controlled to E2 = 1 154700539, 0, 3 622008471as (a) α = 0 95, k1 = 8 5, k2 = 1 5, and k3 = 50; (b) α = 1 1, k1 = 8 5, k2 = 250, and k3 = 50.

11Complexity

So, systems (39) and (40) are synchronized for α = 1 1, k1 =15, k2 = 0 1, k3 = 10 as shown in Figure 12.

9. Concluding Remarks

In this paper, nonlinear dynamics, conditions of chaos con-trol, and synchronization are studied in the fractionalSamardzija-Greller population model with fractional orderbetween zero and two. To the best of the authors’ knowl-edge, dynamical behaviors and chaos applications in thismodel have not been investigated elsewhere when the

fractional order lies between zero and two. This kind ofstudy has great importance to predator-prey ecosystemssince it provides greater degrees of freedom in modelingsuch systems. In addition, increasing the interval of frac-tional parameter helps to raise the complexity in the systemsince the fractional parameter is in a close relationship withfractals which are abundant in ecosystems. Furthermore, asthe fractional parameter α is increased, more accuratedescription of the whole time domain of the process isachieved and the system will show rich dynamics such asthe existence of chaos.

1

e1e2e3

2 3 4 5 6 7 8 9 100t

−0.1

−0.05

0

0.05

0.1

0.15

0.2Sy

nchr

oniz

atio

n er

rors

(a)

e1e2e3

−0.05

0

0.05

0.1

0.15

0.2

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100t

(b)

Figure 10: The synchronization errors (41) converge to zero as using the parameter values a = 0 8 and b = c = 3, the controllers (43), the FCGsk1 = k2 = k3 = 20, and fractional order (a) α = 0 99 and (b) α = 1 1.

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Sync

hron

izat

ion

erro

rs

5 10 15 20 25 35 35 400t

e1e2e3

(a)

10 20 30 40 50 60 70 80 90 1000t

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Sync

hron

izat

ion

erro

rs

e1e2e3

(b)

Figure 11: The synchronization errors (41) with a = 3, b = 4, and c = 10 and the controllers (47) with FCGs k1 = 3 9, k2 = 0 1, and k3 = 10 (a)converge to zero when α = 0 95 and (b) oscillate about zero when α = 1 0.

12 Complexity

By using the fractional Routh-Hurwitz criterion, localstability in this model has been demonstrated as α ∈ 0, 2 .Moreover, the chaoticity of this model has been numericallyexamined by calculating the maximal Lyapunov exponentsusing the Wolf’s algorithm. The calculations show that chaosexists in this system when the fractional parameter lies in theintervals (0, 1) and (1, 2).

Chaos control in this system has been achieved via anovel linear control technique as α ∈ 1, 2 . Furthermore,the role of fractional parameter α on controlling chaos in thismodel has also been illustrated using a linear feedback con-trol technique. It has been proved that using the appropriateFCGs, the trajectory of Samardzija-Greller population modelis stabilized to the steady state E1 = 1, 1 + c, 0 as the frac-tional parameter α lies between zero and one.

Chaos synchronization has been achieved in this systemvia nonlinear feedback control method. It has also beenshown that, under certain choice of nonlinear controllers,the fractional-order Samardzija-Greller model is synchro-nized only when the fractional parameter α is less than one.Thus, the obtained results illustrate that the parameter αhas also a stabilizing role on the synchronization process inthis system.

All the analytical results have been verified using numer-ical simulations.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that this article content has no conflictof interest.

Acknowledgments

This work is supported by the Deanship of Scientific Researchat King Saud University, Research group no. RG-1438-046.

References

[1] H. Zhang, D. Chen, B. Xu, and F. Wang, “Nonlinear modelingand dynamic analysis of hydro-turbine governing system inthe process of load rejection transient,” Energy Conversionand Management, vol. 90, pp. 128–137, 2015.

[2] H. Zhang, D. Chen, C.Wu, X.Wang, J.-M. Lee, and K.-H. Jung,“Dynamic modeling and dynamical analysis of pump-turbinesin S-shaped regions during runaway operation,” Energy Con-version and Management, vol. 138, pp. 375–382, 2017.

[3] H. Zhang, D. Chen, C. Wu, and X. Wang, “Dynamics analysisof the fast-slow hydro-turbine governing system with differenttime-scale coupling,” Communications in Nonlinear Scienceand Numerical Simulation, vol. 54, pp. 136–147, 2018.

[4] H. Zhang, D. Chen, B. Xu, E. Patelli, and S. Tolo, “Dynamicanalysis of a pumped-storage hydropower plant with randompower load,” Mechanical Systems and Signal Processing,vol. 100, pp. 524–533, 2018.

[5] H. Zhang, D. Chen, P. Guo, X. Luo, and A. George, “A novelsurface-cluster approach towards transient modeling ofhydro-turbine governing systems in the start-up process,”Energy Conversion and Management, vol. 165, pp. 861–868,2018.

[6] A. S. Hegazi and A. E. Matouk, “Dynamical behaviors andsynchronization in the fractional order hyperchaotic Chensystem,” Applied Mathematics Letters, vol. 24, no. 11,pp. 1938–1944, 2011.

[7] A. E. Matouk, “On the periodic orbits bifurcating from a foldHopf bifurcation in two hyperchaotic systems,” Optik,vol. 126, no. 24, pp. 4890–4895, 2015.

[8] S. Lv andM. Zhao, “The dynamic complexity of a three speciesfood chain model,” Chaos, Solitons & Fractals, vol. 37, no. 5,pp. 1469–1480, 2008.

[9] M. Javidi and N. Nyamoradi, “Dynamic analysis of a fractionalorder prey-predator interaction with harvesting,” AppliedMathematical Modelling, vol. 37, no. 20-21, pp. 8946–8956,2013.

[10] A. E. Matouk and A. A. Elsadany, “Dynamical analysis, stabi-lization and discretization of a chaotic fractional-order GLVmodel,” Nonlinear Dynamics, vol. 85, no. 3, pp. 1597–1612,2016.

[11] E. N. Lorenz, “Deterministic non-periodic flow,” Journal of theAtmospheric Sciences, vol. 20, no. 2, pp. 130–141, 1963.

[12] H. N. Agiza and A. E. Matouk, “Adaptive synchronization ofChua’s circuits with fully unknown parameters,” Chaos,Solitons & Fractals, vol. 28, no. 1, pp. 219–227, 2006.

[13] A. E. Matouk and H. N. Agiza, “Bifurcations, chaos and syn-chronization in ADVP circuit with parallel resistor,” Journalof Mathematical Analysis and Applications, vol. 341, no. 1,pp. 259–269, 2008.

[14] D. Xiao, J. An, Y. He, and M. Wu, “The chaotic characteristicof the carbon-monoxide utilization ratio in the blast furnace,”ISA Transactions, vol. 68, pp. 109–115, 2017.

[15] R. Luo, H. Su, and Y. Zeng, “Chaos control and synchroniza-tion via switched output control strategy,” Complexity,vol. 2017, Article ID 6125102, 11 pages, 2017.

0.2

0.15

0.1

0.05

Sync

hron

izat

ion

erro

rs

0

–0.05

–0.15

e1

e2

e3

10 15 20t

25 30 35 400

Figure 12: The synchronization errors (41) converge to zero asusing the parameters a = 3, b = 4, and c = 10, fractional order α =1 1, and controllers (47) with FCGs k1 = 15, k2 = 0 1, and k3 = 10.

13Complexity

[16] H. Poincaré, The Foundations of Science: Science and Hypoth-esis, The Value of Science, Science and Methods, The SciencePress, 1913.

[17] I. Podlubny, Fractional Differential Equations, AcademicPress, New York, NY, USA, 1990.

[18] A. E. Matouk, “Dynamical behaviors, linear feedback controland synchronization of the fractional order Liu system,” Jour-nal of Nonlinear Systems and Applications, vol. 1, pp. 135–140,2010.

[19] A. S. Hegazi, E. Ahmed, and A. E. Matouk, “The effect of frac-tional order on synchronization of two fractional order chaoticand hyperchaotic systems,” Journal of Fractional Calculus andApplications, vol. 1, no. 3, pp. 1–15, 2011.

[20] A. E. Matouk and A. A. Elsadany, “Achieving synchronizationbetween the fractional-order hyperchaotic novel and Chensystems via a new nonlinear control technique,” AppliedMath-ematics Letters, vol. 29, pp. 30–35, 2014.

[21] A. E. Matouk, “Chaos synchronization of a fractional-ordermodified Van der Pol-Duffing system via new linear control,backstepping control and Takagi-Sugeno fuzzy approaches,”Complexity, vol. 21, Supplement 1124 pages, 2016.

[22] A. M. A. El-Sayed, H. M. Nour, A. Elsaid, A. E. Matouk, andA. Elsonbaty, “Dynamical behaviors, circuit realization, chaoscontrol and synchronization of a new fractional orderhyperchaotic system,” Applied Mathematical Modelling,vol. 40, no. 5-6, pp. 3516–3534, 2016.

[23] A. E. Matouk, “Chaos, feedback control and synchronizationof a fractional-order modified autonomous Van der Pol-Duffing circuit,” Communications in Nonlinear Science andNumerical Simulation, vol. 16, no. 2, pp. 975–986, 2011.

[24] R. Hilfer, Applications of Fractional Calculus in Physics, WorldScientific, New Jersey, 2000.

[25] A. A. Elsadany and A. E. Matouk, “Dynamical behaviors offractional-order Lotka-Volterra predator-prey model and itsdiscretization,” Journal of Applied Mathematics and Comput-ing, vol. 49, no. 1-2, pp. 269–283, 2015.

[26] A. E. Matouk, A. A. Elsadany, E. Ahmed, and H. N. Agiza,“Dynamical behavior of fractional-order Hastings-Powell foodchain model and its discretization,” Communications in Non-linear Science and Numerical Simulation, vol. 27, no. 1-3,pp. 153–167, 2015.

[27] A. M. A. El-Sayed, A. Elsonbaty, A. A. Elsadany, and A. E.Matouk, “Dynamical analysis and circuit simulation of a newfractional-order hyperchaotic system and its discretization,”International Journal of Bifurcation and Chaos, vol. 26,no. 13, article 1650222, p. 35, 2016.

[28] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys-ical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990.

[29] L. M. Pecora and T. L. Carroll, “Synchronization in chaoticsystems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824,1990.

[30] M. P. Aghababa, “Control of non-integer-order dynamical sys-tems using sliding mode scheme,” Complexity, vol. 21, no. 6,233 pages, 2016.

[31] P. Muthukumar, P. Balasubramaniam, and K. Ratnavelu,“Sliding mode control for generalized robust synchronizationof mismatched fractional order dynamical systems and itsapplication to secure transmission of voice messages,” ISATransactions, 2017, In press.

[32] A. G. Radwan, K. Moaddy, K. N. Salama, S. Momani, andI. Hashim, “Control and switching synchronization of

fractional order chaotic systems using active control tech-nique,” Journal of Advanced Research, vol. 5, no. 1, pp. 125–132, 2014.

[33] N. Samardzija and L. D. Greller, “Explosive route to chaosthrough a fractal torus in a generalized Lotka-Volterra model,”Bulletin of Mathematical Biology, vol. 50, no. 5, pp. 465–491,1988.

[34] J. S. Costello, “Synchronization of chaos in a generalizedLotka–Volterra attractor,” The Nonlinear Journal, vol. 1,pp. 11–17, 1999.

[35] S. Oancea, I. Grosu, and A. V. Oancea, “The pest control inagricultural systems,” Lucrări Științifice, Universitatea deStiinte Agricole Și Medicină Veterinară" Ion Ionescu de laBrad" Iași, Seria Agronomie, vol. 52, no. 1, pp. 218–226,2009, seria Agronomie.

[36] A. A. Elsadany, A. E. Matouk, A. G. Abdelwahab, and H. S.Abdallah, “Dynamical analysis, linear feedback control andsynchronization of a generalized Lotka–Volterra system,”International Journal of Dynamics and Control, vol. 6, no. 1,pp. 328–338, 2018.

[37] M. Srivastava, S. K. Agrawal, and S. Das, “Synchronization ofchaotic fractional order Lotka–Volterra system,” InternationalJournal of Nonlinear Science, vol. 13, pp. 482–494, 2012.

[38] S. K. Agrawal, M. Srivastava, and S. Das, “Synchronizationbetween fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems,” Nonlinear Dynamics, vol. 69, no. 4,pp. 2277–2288, 2012.

[39] E. Ahmed and A. S. Elgazzar, “On fractional order differentialequations model for nonlocal epidemics,” Physica A, vol. 379,no. 2, pp. 607–614, 2007.

[40] D. Matignon, “Stability results for fractional differentialequations with applications to control processing,” in Compu-tational Engineering in Systems and Application Multiconfer-ence, Vol. 2, pp. 963–968, IMACS, IEEE-SMC Proceedings,Lille, 1996.

[41] A. E. Matouk, “Stability conditions, hyperchaos and control ina novel fractional order hyperchaotic system,” Physics LettersA, vol. 373, no. 25, pp. 2166–2173, 2009.

[42] R. Zhang, G. Tian, S. Yang, and H. Cao, “Stability analysis of aclass of fractional order nonlinear systems with order lying in(0, 2),” ISA Transactions, vol. 56, pp. 102–110, 2015.

[43] M. S. Tavazoei and M. Haeri, “A proof for non existence ofperiodic solutions in time invariant fractional order systems,”Automatica, vol. 45, no. 8, pp. 1886–1890, 2009.

[44] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Deter-mining Lyapunov exponents from a time series,” Physica D:Nonlinear Phenomena, vol. 16, no. 3, pp. 285–317, 1985.

14 Complexity

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