Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 397504 13 pageshttpdxdoiorg1011552013397504

Research ArticleSliding Mode Control of the Fractional-OrderUnified Chaotic System

Jian Yuan Bao Shi Xiaoyun Zeng Wenqiang Ji and Tetie Pan

Institute of Systems Science and Mathematics Naval Aeronautical and Astronautical University Yantai Shandong 264001 China

Correspondence should be addressed to Jian Yuan yuanjianscargmailcom

Received 26 June 2013 Revised 9 September 2013 Accepted 9 September 2013

Academic Editor Dumitru Baleanu

Copyright copy 2013 Jian Yuan et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with robust synchronization of the fractional-order unified chaotic systems Firstly control design forsynchronization of nominal systems is proposed via fractional sliding mode techniqueThen systematic uncertainties and externaldisturbances are considered in the fractional-order unified chaotic systems and adaptive sliding mode control is designed forthe synchronization issue Finally numerical simulations are carried out to verify the effectiveness of the two proposed controltechniques

1 Introduction

Even though the theory of fractional calculus dates back tothe end of the 17th century the subject only really cameto life over the last few decades [1] The most significantadvantage of fractional calculus is that it provides a powerfulinstrument of describing memory and hereditary propertiesof different substances [2] In particular fractional differentialequations as the basic theory for fractional-order control [3]have become a powerful tool in describing the dynamics ofcomplex systems and gained great development very recently[4ndash6]

One of the most important areas of application is thefractional-order chaotic systems which have wide potentialapplications in engineering Since Hartley et al firstly dis-covered chaotic phenomenon in fractional dynamics systems[7] there has emerged great interest in this novel andpromising topic On one hand more and more fractionalnonlinear systems which exhibit chaos have been discov-ered and their chaotic behaviors have been studied withnumerical simulations such as the fractional-order Chuacircuit [8] the fractional-order Van der Pol oscillator [9ndash11]the fractional-order Lorenz system [12 13] the fractional-order Chen system [14ndash16] the fractional-order Lu system[17] the fractional-order Liu system [18] the fractional-order

Rossler system [19 20] the fractional-order Arneodo system[21] the fractional-order Lotka-Volterra system [22 23] thefractional-order financial system [24 25] and the discretefractional logistic map [26] On the other hand control andsynchronization of fractional-order dynamical systems havebeen attracting growing investigations Linear-state feedbackcontrol approach has been designed in [14 27ndash35] nonlinearfeedback control in [36ndash40] fractional PID control in [4142] and open-plus-closed-loop control in [43] To tacklewith modeling inaccuracies and external noises which areunavoidable in the real-world application fractional-ordersliding mode control methodology has been established in[44ndash51]

In this paper we investigate robust synchronization of thefractional-order unified chaotic systems We firstly proposecontrollers to synchronize the nominal systems via fractionalsliding mode technique Secondly we consider systematicuncertainties and external disturbances in the fractional-order unified chaotic systems and establish adaptive slidingmode control for synchronization of the uncertain systems

The rest of this paper is organized as follows Section 2presents some basic definitions and theorems about frac-tional calculus and fractional-order dynamical systemSection 3 describes the general form of fractional-orderunified chaotic system and presents ourmain objective in this

2 Abstract and Applied Analysis

paper Section 4 proposes the sliding mode control designfor synchronization of nominal systems and adaptive slidingmode control design for the uncertain system Numericalsimulations are presented to show the effectiveness of the pro-posed schemes in Section 5 Finally this paper is concludedin Section 6

2 Preliminaries

Definition 1 Themost important function used in fractionalcalculus is Eulerrsquos Gamma function which is defined as

Γ (119899) = int

infin

0

119905119899minus1

119890minus119905

119889119905 (1)

Definition 2 Another important function is a two-parameterfunction of the Mittag-Leffler type defined as

119864120572120573

(119911) =

infin

sum

119896=0

119911119896

Γ (120572119896 + 120573) 120572 gt 0 120573 gt 0 (2)

Fractional calculus is a generalization of integrationand differentiation to noninteger-order fundamental opera-tor119886119863120572

119905 where 119886 and 119905 are the bounds of the operation and

119886 isin R The continuous integrodifferential operator is definedas

119886119863120572

119905=

119889120572

119889119905120572 120572 gt 0

1 120572 = 0

int

119905

119886

(119889120591)120572

120572 lt 0

(3)

The threemost frequently used definitions for the generalfractional calculus are the Grunwald-Letnikov definition theRiemann-Liouville definition and the Caputo definition [223 52 53]

Definition 3 TheGrunwald-Letnikov derivative definition oforder 120572 is described as

119886119863120572

119905119891 (119905) = lim

ℎrarr0

1

ℎ120572

infin

sum

119895=0

(minus1)119895

(120572

119895)119891 (119905 minus 119895ℎ) (4)

For binomial coefficients calculation we can use the rela-tion between Eulerrsquos Gamma function and factorial definedas

(120572

119895) =

120572

119895 (120572 minus 119895)=

Γ (120572 + 1)

Γ (119895 + 1) Γ (120572 minus 119895 + 1)(5)

for

(120572

0) = 1 (6)

Definition 4 The Riemann-Liouville derivative definition oforder 120572 is described as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

119886

119891 (120591) 119889120591

(119905 minus 120591)120572minus119899+1

119899 minus 1 lt 120572 lt 119899

(7)

However applied problems require definitions of frac-tional derivatives allowing the utilization of physically inter-pretable initial conditions which contain 119891(119886) 1198911015840(119886) andso forth Unfortunately the Riemann-Liouville approach failsto meet this practical need It is M Caputo who solved thisconflict

Definition 5 The Caputo definition of fractional derivativecan be written as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)int

119905

119886

119891(119899)

(120591) 119889120591

(119905 minus 120591)120572minus119899+1

119899 minus 1 lt 120572 lt 119899 (8)

In the following we use the Caputo approach to describethe fractional chaotic systems and the Grunwald-Letnikovapproach to propose numerical simulations To simplify thenotation we denote the fractional-order derivative as 119863

120572

instead of0119863120572

119905in this paper

Lemma 6 (see [22]) Consider the following commensuratefractional-order dynamics system

119863120572

119909 = 119891 (119909) (9)

where 0 lt 120572 le 1 and 119909 isin R119899 The equilibrium points of system(9) are calculated by solving the following equation

119891 (119909) = 0 (10)

These points are locally asymptotically stable if all eigenvalues120582119894of the Jacobian matrix 119869 = 120597119891120597119909 evaluated at the

equilibrium points satisfy

1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

120572120587

2 (11)

Lemma 7 (see [22]) Consider the following 119899-dimensionallinear fractional-order dynamics system

11986311990211199091= 119886111199091+ 119886121199092+ sdot sdot sdot + 119886

1119899119909119899

11986311990221199092= 119886211199091+ 119886221199092+ sdot sdot sdot + 119886

2119899119909119899

sdot sdot sdot

119863119902119899119909119899= 11988611989911199091+ 11988611989921199092+ sdot sdot sdot + 119886

119899119899119909119899

(12)

where all 120572119894rsquos are rational numbers between 0 and 1 Assume

119872 be the lowest common multiple of the denominators 119906119894rsquos of

120572119894rsquos where 120572

119894= V119894119906119894 (119906119894 V119894) = 1 and 119906

119894 V119894isin 119885+ for 119894 =

1 2 119899 Define

Δ (120582) = (

1205821198721205721 minus 11988611

minus11988612

sdot sdot sdot minus1198861119899

minus11988621

1205821198721205722 minus 11988622

sdot sdot sdot minus1198862119899

d

minus1198861198991

minus1198861198992

sdot sdot sdot 120582119872120572119899 minus 119886119899119899

) (13)

Then the zero solution of system (12) is globally asymptot-ically stable in the Lyapunov sense if all roots of the equationdet(Δ(120582)) = 0 satisfy | arg(120582)| gt 1205872119872

Abstract and Applied Analysis 3

Lemma8 (see [53]) Assume that there exists a scalar function119881 of the state 119909with continuous first-order derivative such thatthe following are given

(i) 119881(119909) is positive definite(ii) (119909) is negative definite(iii) 119881(119909) rarr infin as 119909 rarr infin

Then the equilibrium at the origin is globally asymptoticallystable

3 Problem Formulation

In [54] Lu et al have considered a kind of chaotic systems andpointed out that these systems can be described in a unifiedform as follows

1= (25120572 + 10) (119909

2minus 1199091)

2= (28 minus 35120572) 119909

1minus 11990911199093+ (29120572 minus 1) 119909

2

3= 11990911199092minus

(8 + 120572) 1199093

3

(14)

where 1199091 1199092 and 119909

3are state variables and 120572 isin [0 1] is

the system parameter Lu et al [54] call system (14) a unifiedchaotic system because it is chaotic for any 120572 isin [0 1] When120572 isin [0 08) system (14) is called a the generalized Lorenzchaotic system When 120572 = 08 it is called the Lu chaoticsystem And it is called the generalized Chen chaotic systemwhen 120572 isin (08 1]

The fractional-order unified chaotic system has beenfirstly introduced and studied in [55] and reads as

11986311990211199091= (25120572 + 10) (119909

2minus 1199091)

11986311990221199092= (28 minus 35120572) 119909

1minus 11990911199093+ (29120572 minus 1) 119909

2

11986311990231199093= 11990911199092minus

(8 + 120572) 1199093

3

(15)

where 1199021 1199022 1199023isin (0 1] is the fractional order

System (15) is considered as the drive (master) system andthe response (slave) system is a controlled system as follows

11986311990211199101= (25120572 + 10) (119910

2minus 1199101) + 1199061

11986311990221199102= (28 minus 35120572) 119910

1minus 11991011199103+ (29120572 minus 1) 119910

2+ 1199062

11986311990231199103= 11991011199102minus

(8 + 120572) 1199103

3+ 1199063

(16)

Let us define the state errors between the response system(16) and the drive system (15) as 119890

1= 1199101minus 1199091 1198902= 1199102minus 1199092

and 1198903= 1199103minus 1199093

By subtracting (15) from (16) one can get the followingerror dynamical system

11986311990211198901= (25120572 + 10) (119890

2minus 1198901) + 1199061

11986311990221198902= (28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103

minus 11989031199101+ (29120572 minus 1) 119890

2+ 1199062

11986311990231198903= minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ 1199063

(17)

Ourmain objective in this paper is to investigate the synchro-nization issue for the fractional-order unified chaotic system(15) It is clear that the synchronization of systems (15) and(16) is equivalent to the stabilization of the error dynamicalsystem (17)

4 Synchronization Design

In the following the sliding mode control technique whichcan maintain low sensitivity to unmodeled dynamics andexternal disturbances is applied to establish an effectivecontrol law to guarantee the synchronization of the drivesystem (15) and the response system (16) Two major stepsare involved in the sliding mode control design firstlyconstructing an appropriate sliding surface on which thedesired system dynamics is stable and secondly developing asuitable control law such that the sliding condition is attained

41 Synchronization of the Nominal System In this sub-section let us firstly consider a simple case the nominalfractional-order unified chaotic system that is the systemcontains no systematic uncertainties or external disturbancesThe design procedure is elaborated in the rest part of thissubsection

411 Sliding Surfaces Design In order to achieve the stabilityof system (17) three sliding surfaces 119878

1 1198782 and 119878

3are

introduced as

119904119894(119905) = (119863

119902119894 + 120582119894) int

119905

0

119890119894(120591) 119889120591 119894 = 1 2 3 (18)

the time derivative of which becomes

119904119894(119905) = 119863

119902119894119890119894(119905) + 120582

119894119890119894(119905) 119894 = 1 2 3 (19)

As long as system (17) operates on the sliding surfacesit satisfies 119904

119894= 0 and 119904

119894= 0 119894 = 1 2 3 which yields the

following sliding mode dynamics

11986311990211198901= minus12058211198901

11986311990221198902= minus12058221198902

11986311990231198903= minus12058231198903

(20)

By using Lemmas 6 or 7 system (20) is asymptoticallystable As a result the sliding mode surfaces (18) we have justconstructed are appropriate for the control design

412 Control Laws Design

Step 1 Choose the first control Lyapunov function

1198811(119905) =

1

21199042

1 (21)

Taking time derivative gives

1(119905) = 119904

11199041= 1199041(11986311990211198901+ 12058211198901) (22)

4 Abstract and Applied Analysis

Substituting the first state equation of (17) into (22) one has

1(119905) = 119904

1[(25120572 + 10) (119890

2minus 1198901) + 1199061+ 12058211198901] (23)

Therefore by designing the first control law as

1199061= minus (25120572 + 10) (119890

2minus 1198901) minus 12058211198901minus 1198961sgn (119904

1) (24)

where 1198961gt 0 and

sgn (119904) =

1 119904 gt 0

0 119904 = 0

minus1 119904 lt 0

(25)

then (23) becomes

1(119905) = minus119896

1

100381610038161003816100381611990411003816100381610038161003816

(26)

Step 2 Choose the second control Lyapunov function

1198812(119905) =

1

21199042

2 (27)

By taking its derivative with respect to time yields

2(119905) = 119904

21199042= 1199042(11986311990221198902+ 12058221198902) (28)

Substituting the second state equation of (17) into (28) onehas

2(119905) = 119904

2[(28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103minus 11989031199101

+ (29120572 minus 1) 1198902+ 1199062+ 12058221198902]

(29)

Therefore by designing the second control law as

1199062= minus (28 minus 35120572) 119890

1minus 11989011198903+ 11989011199103+ 11989031199101

minus (29120572 minus 1) 1198902minus 12058221198902minus 1198962sgn (119904

3)

(30)

where 1198962gt 0

Equation (29) becomes

2(119905) = minus119896

2

100381610038161003816100381611990421003816100381610038161003816

(31)

Step 3 Choose the third control Lyapunov function

1198813(119905) =

1

21199042

3 (32)

Its time derivative is given by

3(119905) = 119904

31199043= 1199043(11986311990231198903+ 12058231198903) (33)

Substituting the third state equation of (17) into (33) one has

3(119905) = 119904

3[minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ 1199063+ 12058231198903]

(34)

We are then in the position to design the third control lawas follows

1199063= 11989011198902minus 11989011199102minus 11989021199101+

(8 + 120572) 1198903

3minus 12058231198903minus 1198963sgn (119904

3)

(35)

where 1198963gt 0

With this choice (34) can be rewritten as

3(119905) = minus119896

3

100381610038161003816100381611990431003816100381610038161003816

(36)

Step 4 Finally we gather the above three control functions as

119881 (1199041 1199042 1199043) =

1

21199042

1+

1

21199042

2+

1

21199042

3 (37)

It is clear from (26) (31) and (36) that

(1199041 1199042 1199043) = minus (119896

1

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(38)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (39)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (40)

The resulting derivative of 119881 is

(1199041 1199042 1199043) lt minus119896 119904 (41)

In terms of Lemma 8 the Lyapunov function (37) pro-vides the proof of globally asymptotical stability with thecontrol laws (24) (30) and (35)

42 Synchronization of the Uncertain System In this subsec-tion we will proceed to study the synchronization of thefractional-order unified chaotic system in the presence ofsystematic uncertainties and external disturbances which canbe hardly ignored in the real-world application It is assumedthat systematic uncertainties Δ119891

1 Δ1198912 and Δ119891

3and external

disturbances 1198891 1198892 and 119889

3are all bounded that is |Δ119891

119894| lt

120588119894and |119889

119894(119905)| lt 120579

119894 where 120588

119894and 120579

119894are unknown positive

constants 119894 = 1 2 3 Let us denote that 120588119894is an estimate of 120588

119894

while 120579119894is an estimate of 120579

119894 Since 120588

119894and 120579119894are unknown our

task in this subsection is fulfilled with an adaptive controllerconsisting of control laws and update laws to obtain 120588

119894and 120579119894

119894 = 1 2 3The uncertain fractional-order unified chaotic system can

be described as

11986311990211198901= (25120572 + 10) (119890

2minus 1198901) + Δ119891

1+ 1198891+ 1199061

11986311990221198902= (28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103minus 11989031199101

+ (29120572 minus 1) 1198902+ Δ1198912+ 1198892+ 1199062

11986311990231198903= minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ Δ1198913+ 1198893+ 1199063

(42)

The adaptive sliding mode design of system (42) consistsof four steps which are elaborated as follows

Step 1 Consider the first control Lyapunov function

1198811(119905) =

1

2[1199042

1+

1

1205831

(1205881minus 1205881)2

+1

1205741

(1205791minus 1205791)2

] (43)

Abstract and Applied Analysis 5

the time derivative of which becomes

1(119905) = 119904

11199041+

1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

= 1199041(11986311990211198901+ 12058211198901) +

1

1205831

(1205881minus 1205881) 1205881

+1

1205741

(1205791minus 1205791)

1205791

(44)

Substituting the first state equation of (17) into (44) one has

1(119905) = 119904

1[(25120572 + 10) (119890

2minus 1198901) + Δ119891

1+ 1198891+ 1199061+ 12058211198901]

+1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

(45)

By designing the first control law and adaptive law as

1199061= minus (25120572 + 10) (119890

2minus 1198901) minus 12058211198901minus (1205881+ 1205791+ 1198961) sgn (119904

1)

(46)

1205881= 1205831

100381610038161003816100381611990411003816100381610038161003816

1205791= 1205741

100381610038161003816100381611990411003816100381610038161003816

(47)

then (45) becomes

1= 119904 [(Δ119891

1+ 1198891) minus (120588

1+ 1205791+ 1198961) sgn (119904

1)]

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

le (1003816100381610038161003816Δ1198911

1003816100381610038161003816 +10038161003816100381610038161198891

1003816100381610038161003816)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

lt (1205881+ 1205791)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816 = minus1198961

100381610038161003816100381611990411003816100381610038161003816

(48)

Step 2 Choose the second control Lyapunov function

1198812(119905) =

1

2[1199042

2+

1

1205832

(1205882minus 1205882)2

+1

1205742

(1205792minus 1205792)2

] (49)

whose derivative is

2(119905) = 119904

21199042+

1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

= 1199042(11986311990221198902+ 12058221198902) +

1

1205832

(1205882minus 1205882) 1205882

+1

1205741

(1205792minus 1205792)

1205792

(50)

Substituting the second state equation of (17) into (50) onehas

2(119905) = 119904

2[(28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103

minus11989031199101+ (29120572 minus 1) 119890

2]

+ 1199042[Δ1198912+ 1198892+ 1199062+ 12058221198902]

+1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

(51)

We choose the control law and the adaptive law

1199062= minus (28 minus 35120572) 119890

1minus 11989011198903+ 11989011199103+ 11989031199101

minus (29120572 minus 1) 1198902minus 12058221198902minus (1205882+ 1205792+ 1198962) sgn (119904

2)

(52)

1205882= 1205832

100381610038161003816100381611990421003816100381610038161003816

(53)

1205792= 1205742

100381610038161003816100381611990421003816100381610038161003816

(54)

With this choice (51) becomes

2(119905) lt minus119896

2

100381610038161003816100381611990421003816100381610038161003816

(55)

Step 3 Choose the third Lyapunov function

1198813(119905) =

1

2[1199042

3+

1

1205833

(1205883minus 1205883)2

+1

1205743

(1205793minus 1205793)2

] (56)

Taking time derivative gives

3(119905) = 119904

31199043+

1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

= 1199043(11986311990231198903+ 12058231198903) +

1

1205833

(1205883minus 1205883) 1205883

+1

1205743

(1205793minus 1205793)

1205793

(57)

Substituting the third state equation of (17) into (57) one has

3(119905) = 119904

3[minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3]

+ 1199043[Δ1198913+ 1198893+ 1199063+ 12058231198903]

+1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

(58)

We choose the third control law and adaptive law

1199063= 11989011198902minus 11989011199102minus 11989021199101+

(8 + 120572) 1198903

3minus 12058231198903

minus (1205883+ 1205793+ 1198963) sgn (119904

3)

(59)

1205883= 1205833

100381610038161003816100381611990431003816100381610038161003816

1205793= 1205743

100381610038161003816100381611990431003816100381610038161003816

(60)

Then the resulting derivative of 1198813is

3(119905) lt minus119896

3

100381610038161003816100381611990431003816100381610038161003816

(61)

6 Abstract and Applied Analysis

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus30

minus20

minus10

0

10

20

Time (s)

x(t)

(b)

0 50 100minus40

minus20

minus10

0

10

20

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 1 Chaotic trajectories with 120572 = 05

Step 4 Finally we gather the above three control functions as

119881 (119905) =1

2

3

sum

119894=1

1199042

119894+

3

sum

119894=1

1

2120583119894

(120588119894minus 120588119894)2

+

3

sum

119894=1

1

2120574119894

(120579119894minus 120579119894)2

(62)

It is clear from (48) (55) and (61) that

(119905) = minus (1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(63)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (64)

The resulting derivative of is

(1199041 1199042 1199043) lt minus119896 119904 (65)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (66)

By using Lemma 8 this Lyapunov function provides theproof of globally asymptotical stability with the control laws(46) (52) and (59) and adaptive laws (47) (53) and (60)

5 Numerical Simulations

51 Chaotic Behaviors of Fractional-Order Chaotic System In[55] the authors have provided us with numerical methodsof fractional calculus In [56] the chaotic behaviors of thefractional-order unified system were numerically investi-gated where it is found that the lowest order to exhibit chaosis 276

The chaotic behaviors are presented in Figures 1 and2 with fractional orders of 119902

1= 093 119902

2= 094 and

Abstract and Applied Analysis 7

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus40

minus20

0

20

40

Time (s)

x(t)

(b)

0 50 100minus40

minus20

0

20

40

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 2 Chaotic trajectories with 120572 = 1

1199023= 095 and the initial conditions of (119909

1(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1)Thenumerical algorithm is based on the followingGrunwald-Letnikovrsquos definition

119909 (119905119896) = (25120572 + 10) (119909

2(119905119896minus1

) minus 1199091(119905119896minus1

)) ℎ1199021

minus

119896

sum

119895=2

119888(1199021)

1198951199091(119905119896minus119895

)

1199092(119905119896) = [(28 minus 35120572) 119909

1(119905119896) minus 1199091(119905119896) 1199093(119905119896minus1

)

+ (29120572 minus 1) 1199092(119905119896minus1

)] ℎ1199022 minus

119896

sum

119895=2

119888(1199022)

1198951199092(119905119896minus119895

)

1199093(119905119896) = [119909

1(119905119896) 1199092(119905119896) minus

120572 + 8

31199093(119905119896minus1

)] ℎ1199023

minus

119896

sum

119895=2

119888(1199023)

1198951199093(119905119896minus119895

)

(67)

where 119879sim is the simulation time 119896 = 1 2 119873 for 119873 =

[119879simℎ]

52 Simulations of Synchronization of the Nominal SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed sliding modelcontrol in Section 41 In the numerical simulations thefractional orders are chosen as 119902

1= 093 119902

2= 094 and

1199023= 095 The initial conditions of the drive system (15) and

the response system (16) are chosen as (1199091(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1) and (1199101(0) 1199102(0) 1199103(0)) = (4 minus4 4) respectively

Parameters in (18) are chosen as 1205821= 1205822= 1205823= 4 Gains

of the control inputs in (24) (30) and (35) are chosen as1198961= 1198962= 1198962= 1

When 120572 = 05 and 120572 = 1 numerical simulations ofsynchronization of system (15) are presented in Figures 34 5 6 7 and 8 with control inputs (24) (30) and (35)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

8 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 3 Synchronization of determined fractional-order unified chaotic system with 120572 = 05 (blue line represents the trajectories of thedrive system while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 4 Synchronization errors states with 120572 = 05 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

0 2 4 6 8 10

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 5 Sliding surfaces states with 120572 = 05 (red line represents the first sliding surface state 1199041 blue line represents the second sliding

surface state 1199042 and black line represents the third sliding surface state 119904

3)

Abstract and Applied Analysis 9

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 6 Synchronization of determined fractional-order unified chaotic system with 120572 = 1 (blue line represents the trajectories of the drivesystem while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus2

minus1

Figure 7 Synchronization errors states with 120572 = 1 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

2 4 6 8 10

0

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 8 Sliding surfaces states with 120572 = 1 (red line represents the first sliding surface state 1199041 blue line represents the second sliding surface

state 1199042 and black line represents the third sliding surface state 119904

3)

10 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

paper Section 4 proposes the sliding mode control designfor synchronization of nominal systems and adaptive slidingmode control design for the uncertain system Numericalsimulations are presented to show the effectiveness of the pro-posed schemes in Section 5 Finally this paper is concludedin Section 6

2 Preliminaries

Definition 1 Themost important function used in fractionalcalculus is Eulerrsquos Gamma function which is defined as

Γ (119899) = int

infin

0

119905119899minus1

119890minus119905

119889119905 (1)

Definition 2 Another important function is a two-parameterfunction of the Mittag-Leffler type defined as

119864120572120573

(119911) =

infin

sum

119896=0

119911119896

Γ (120572119896 + 120573) 120572 gt 0 120573 gt 0 (2)

Fractional calculus is a generalization of integrationand differentiation to noninteger-order fundamental opera-tor119886119863120572

119905 where 119886 and 119905 are the bounds of the operation and

119886 isin R The continuous integrodifferential operator is definedas

119886119863120572

119905=

119889120572

119889119905120572 120572 gt 0

1 120572 = 0

int

119905

119886

(119889120591)120572

120572 lt 0

(3)

The threemost frequently used definitions for the generalfractional calculus are the Grunwald-Letnikov definition theRiemann-Liouville definition and the Caputo definition [223 52 53]

Definition 3 TheGrunwald-Letnikov derivative definition oforder 120572 is described as

119886119863120572

119905119891 (119905) = lim

ℎrarr0

1

ℎ120572

infin

sum

119895=0

(minus1)119895

(120572

119895)119891 (119905 minus 119895ℎ) (4)

For binomial coefficients calculation we can use the rela-tion between Eulerrsquos Gamma function and factorial definedas

(120572

119895) =

120572

119895 (120572 minus 119895)=

Γ (120572 + 1)

Γ (119895 + 1) Γ (120572 minus 119895 + 1)(5)

for

(120572

0) = 1 (6)

Definition 4 The Riemann-Liouville derivative definition oforder 120572 is described as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

119886

119891 (120591) 119889120591

(119905 minus 120591)120572minus119899+1

119899 minus 1 lt 120572 lt 119899

(7)

However applied problems require definitions of frac-tional derivatives allowing the utilization of physically inter-pretable initial conditions which contain 119891(119886) 1198911015840(119886) andso forth Unfortunately the Riemann-Liouville approach failsto meet this practical need It is M Caputo who solved thisconflict

Definition 5 The Caputo definition of fractional derivativecan be written as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)int

119905

119886

119891(119899)

(120591) 119889120591

(119905 minus 120591)120572minus119899+1

119899 minus 1 lt 120572 lt 119899 (8)

In the following we use the Caputo approach to describethe fractional chaotic systems and the Grunwald-Letnikovapproach to propose numerical simulations To simplify thenotation we denote the fractional-order derivative as 119863

120572

instead of0119863120572

119905in this paper

Lemma 6 (see [22]) Consider the following commensuratefractional-order dynamics system

119863120572

119909 = 119891 (119909) (9)

where 0 lt 120572 le 1 and 119909 isin R119899 The equilibrium points of system(9) are calculated by solving the following equation

119891 (119909) = 0 (10)

These points are locally asymptotically stable if all eigenvalues120582119894of the Jacobian matrix 119869 = 120597119891120597119909 evaluated at the

equilibrium points satisfy

1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

120572120587

2 (11)

Lemma 7 (see [22]) Consider the following 119899-dimensionallinear fractional-order dynamics system

11986311990211199091= 119886111199091+ 119886121199092+ sdot sdot sdot + 119886

1119899119909119899

11986311990221199092= 119886211199091+ 119886221199092+ sdot sdot sdot + 119886

2119899119909119899

sdot sdot sdot

119863119902119899119909119899= 11988611989911199091+ 11988611989921199092+ sdot sdot sdot + 119886

119899119899119909119899

(12)

where all 120572119894rsquos are rational numbers between 0 and 1 Assume

119872 be the lowest common multiple of the denominators 119906119894rsquos of

120572119894rsquos where 120572

119894= V119894119906119894 (119906119894 V119894) = 1 and 119906

119894 V119894isin 119885+ for 119894 =

1 2 119899 Define

Δ (120582) = (

1205821198721205721 minus 11988611

minus11988612

sdot sdot sdot minus1198861119899

minus11988621

1205821198721205722 minus 11988622

sdot sdot sdot minus1198862119899

d

minus1198861198991

minus1198861198992

sdot sdot sdot 120582119872120572119899 minus 119886119899119899

) (13)

Then the zero solution of system (12) is globally asymptot-ically stable in the Lyapunov sense if all roots of the equationdet(Δ(120582)) = 0 satisfy | arg(120582)| gt 1205872119872

Abstract and Applied Analysis 3

Lemma8 (see [53]) Assume that there exists a scalar function119881 of the state 119909with continuous first-order derivative such thatthe following are given

(i) 119881(119909) is positive definite(ii) (119909) is negative definite(iii) 119881(119909) rarr infin as 119909 rarr infin

Then the equilibrium at the origin is globally asymptoticallystable

3 Problem Formulation

In [54] Lu et al have considered a kind of chaotic systems andpointed out that these systems can be described in a unifiedform as follows

1= (25120572 + 10) (119909

2minus 1199091)

2= (28 minus 35120572) 119909

1minus 11990911199093+ (29120572 minus 1) 119909

2

3= 11990911199092minus

(8 + 120572) 1199093

3

(14)

where 1199091 1199092 and 119909

3are state variables and 120572 isin [0 1] is

the system parameter Lu et al [54] call system (14) a unifiedchaotic system because it is chaotic for any 120572 isin [0 1] When120572 isin [0 08) system (14) is called a the generalized Lorenzchaotic system When 120572 = 08 it is called the Lu chaoticsystem And it is called the generalized Chen chaotic systemwhen 120572 isin (08 1]

The fractional-order unified chaotic system has beenfirstly introduced and studied in [55] and reads as

11986311990211199091= (25120572 + 10) (119909

2minus 1199091)

11986311990221199092= (28 minus 35120572) 119909

1minus 11990911199093+ (29120572 minus 1) 119909

2

11986311990231199093= 11990911199092minus

(8 + 120572) 1199093

3

(15)

where 1199021 1199022 1199023isin (0 1] is the fractional order

System (15) is considered as the drive (master) system andthe response (slave) system is a controlled system as follows

11986311990211199101= (25120572 + 10) (119910

2minus 1199101) + 1199061

11986311990221199102= (28 minus 35120572) 119910

1minus 11991011199103+ (29120572 minus 1) 119910

2+ 1199062

11986311990231199103= 11991011199102minus

(8 + 120572) 1199103

3+ 1199063

(16)

Let us define the state errors between the response system(16) and the drive system (15) as 119890

1= 1199101minus 1199091 1198902= 1199102minus 1199092

and 1198903= 1199103minus 1199093

By subtracting (15) from (16) one can get the followingerror dynamical system

11986311990211198901= (25120572 + 10) (119890

2minus 1198901) + 1199061

11986311990221198902= (28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103

minus 11989031199101+ (29120572 minus 1) 119890

2+ 1199062

11986311990231198903= minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ 1199063

(17)

Ourmain objective in this paper is to investigate the synchro-nization issue for the fractional-order unified chaotic system(15) It is clear that the synchronization of systems (15) and(16) is equivalent to the stabilization of the error dynamicalsystem (17)

4 Synchronization Design

In the following the sliding mode control technique whichcan maintain low sensitivity to unmodeled dynamics andexternal disturbances is applied to establish an effectivecontrol law to guarantee the synchronization of the drivesystem (15) and the response system (16) Two major stepsare involved in the sliding mode control design firstlyconstructing an appropriate sliding surface on which thedesired system dynamics is stable and secondly developing asuitable control law such that the sliding condition is attained

41 Synchronization of the Nominal System In this sub-section let us firstly consider a simple case the nominalfractional-order unified chaotic system that is the systemcontains no systematic uncertainties or external disturbancesThe design procedure is elaborated in the rest part of thissubsection

411 Sliding Surfaces Design In order to achieve the stabilityof system (17) three sliding surfaces 119878

1 1198782 and 119878

3are

introduced as

119904119894(119905) = (119863

119902119894 + 120582119894) int

119905

0

119890119894(120591) 119889120591 119894 = 1 2 3 (18)

the time derivative of which becomes

119904119894(119905) = 119863

119902119894119890119894(119905) + 120582

119894119890119894(119905) 119894 = 1 2 3 (19)

As long as system (17) operates on the sliding surfacesit satisfies 119904

119894= 0 and 119904

119894= 0 119894 = 1 2 3 which yields the

following sliding mode dynamics

11986311990211198901= minus12058211198901

11986311990221198902= minus12058221198902

11986311990231198903= minus12058231198903

(20)

By using Lemmas 6 or 7 system (20) is asymptoticallystable As a result the sliding mode surfaces (18) we have justconstructed are appropriate for the control design

412 Control Laws Design

Step 1 Choose the first control Lyapunov function

1198811(119905) =

1

21199042

1 (21)

Taking time derivative gives

1(119905) = 119904

11199041= 1199041(11986311990211198901+ 12058211198901) (22)

4 Abstract and Applied Analysis

Substituting the first state equation of (17) into (22) one has

1(119905) = 119904

1[(25120572 + 10) (119890

2minus 1198901) + 1199061+ 12058211198901] (23)

Therefore by designing the first control law as

1199061= minus (25120572 + 10) (119890

2minus 1198901) minus 12058211198901minus 1198961sgn (119904

1) (24)

where 1198961gt 0 and

sgn (119904) =

1 119904 gt 0

0 119904 = 0

minus1 119904 lt 0

(25)

then (23) becomes

1(119905) = minus119896

1

100381610038161003816100381611990411003816100381610038161003816

(26)

Step 2 Choose the second control Lyapunov function

1198812(119905) =

1

21199042

2 (27)

By taking its derivative with respect to time yields

2(119905) = 119904

21199042= 1199042(11986311990221198902+ 12058221198902) (28)

Substituting the second state equation of (17) into (28) onehas

2(119905) = 119904

2[(28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103minus 11989031199101

+ (29120572 minus 1) 1198902+ 1199062+ 12058221198902]

(29)

Therefore by designing the second control law as

1199062= minus (28 minus 35120572) 119890

1minus 11989011198903+ 11989011199103+ 11989031199101

minus (29120572 minus 1) 1198902minus 12058221198902minus 1198962sgn (119904

3)

(30)

where 1198962gt 0

Equation (29) becomes

2(119905) = minus119896

2

100381610038161003816100381611990421003816100381610038161003816

(31)

Step 3 Choose the third control Lyapunov function

1198813(119905) =

1

21199042

3 (32)

Its time derivative is given by

3(119905) = 119904

31199043= 1199043(11986311990231198903+ 12058231198903) (33)

Substituting the third state equation of (17) into (33) one has

3(119905) = 119904

3[minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ 1199063+ 12058231198903]

(34)

We are then in the position to design the third control lawas follows

1199063= 11989011198902minus 11989011199102minus 11989021199101+

(8 + 120572) 1198903

3minus 12058231198903minus 1198963sgn (119904

3)

(35)

where 1198963gt 0

With this choice (34) can be rewritten as

3(119905) = minus119896

3

100381610038161003816100381611990431003816100381610038161003816

(36)

Step 4 Finally we gather the above three control functions as

119881 (1199041 1199042 1199043) =

1

21199042

1+

1

21199042

2+

1

21199042

3 (37)

It is clear from (26) (31) and (36) that

(1199041 1199042 1199043) = minus (119896

1

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(38)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (39)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (40)

The resulting derivative of 119881 is

(1199041 1199042 1199043) lt minus119896 119904 (41)

In terms of Lemma 8 the Lyapunov function (37) pro-vides the proof of globally asymptotical stability with thecontrol laws (24) (30) and (35)

42 Synchronization of the Uncertain System In this subsec-tion we will proceed to study the synchronization of thefractional-order unified chaotic system in the presence ofsystematic uncertainties and external disturbances which canbe hardly ignored in the real-world application It is assumedthat systematic uncertainties Δ119891

1 Δ1198912 and Δ119891

3and external

disturbances 1198891 1198892 and 119889

3are all bounded that is |Δ119891

119894| lt

120588119894and |119889

119894(119905)| lt 120579

119894 where 120588

119894and 120579

119894are unknown positive

constants 119894 = 1 2 3 Let us denote that 120588119894is an estimate of 120588

119894

while 120579119894is an estimate of 120579

119894 Since 120588

119894and 120579119894are unknown our

task in this subsection is fulfilled with an adaptive controllerconsisting of control laws and update laws to obtain 120588

119894and 120579119894

119894 = 1 2 3The uncertain fractional-order unified chaotic system can

be described as

11986311990211198901= (25120572 + 10) (119890

2minus 1198901) + Δ119891

1+ 1198891+ 1199061

11986311990221198902= (28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103minus 11989031199101

+ (29120572 minus 1) 1198902+ Δ1198912+ 1198892+ 1199062

11986311990231198903= minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ Δ1198913+ 1198893+ 1199063

(42)

The adaptive sliding mode design of system (42) consistsof four steps which are elaborated as follows

Step 1 Consider the first control Lyapunov function

1198811(119905) =

1

2[1199042

1+

1

1205831

(1205881minus 1205881)2

+1

1205741

(1205791minus 1205791)2

] (43)

Abstract and Applied Analysis 5

the time derivative of which becomes

1(119905) = 119904

11199041+

1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

= 1199041(11986311990211198901+ 12058211198901) +

1

1205831

(1205881minus 1205881) 1205881

+1

1205741

(1205791minus 1205791)

1205791

(44)

Substituting the first state equation of (17) into (44) one has

1(119905) = 119904

1[(25120572 + 10) (119890

2minus 1198901) + Δ119891

1+ 1198891+ 1199061+ 12058211198901]

+1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

(45)

By designing the first control law and adaptive law as

1199061= minus (25120572 + 10) (119890

2minus 1198901) minus 12058211198901minus (1205881+ 1205791+ 1198961) sgn (119904

1)

(46)

1205881= 1205831

100381610038161003816100381611990411003816100381610038161003816

1205791= 1205741

100381610038161003816100381611990411003816100381610038161003816

(47)

then (45) becomes

1= 119904 [(Δ119891

1+ 1198891) minus (120588

1+ 1205791+ 1198961) sgn (119904

1)]

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

le (1003816100381610038161003816Δ1198911

1003816100381610038161003816 +10038161003816100381610038161198891

1003816100381610038161003816)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

lt (1205881+ 1205791)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816 = minus1198961

100381610038161003816100381611990411003816100381610038161003816

(48)

Step 2 Choose the second control Lyapunov function

1198812(119905) =

1

2[1199042

2+

1

1205832

(1205882minus 1205882)2

+1

1205742

(1205792minus 1205792)2

] (49)

whose derivative is

2(119905) = 119904

21199042+

1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

= 1199042(11986311990221198902+ 12058221198902) +

1

1205832

(1205882minus 1205882) 1205882

+1

1205741

(1205792minus 1205792)

1205792

(50)

Substituting the second state equation of (17) into (50) onehas

2(119905) = 119904

2[(28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103

minus11989031199101+ (29120572 minus 1) 119890

2]

+ 1199042[Δ1198912+ 1198892+ 1199062+ 12058221198902]

+1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

(51)

We choose the control law and the adaptive law

1199062= minus (28 minus 35120572) 119890

1minus 11989011198903+ 11989011199103+ 11989031199101

minus (29120572 minus 1) 1198902minus 12058221198902minus (1205882+ 1205792+ 1198962) sgn (119904

2)

(52)

1205882= 1205832

100381610038161003816100381611990421003816100381610038161003816

(53)

1205792= 1205742

100381610038161003816100381611990421003816100381610038161003816

(54)

With this choice (51) becomes

2(119905) lt minus119896

2

100381610038161003816100381611990421003816100381610038161003816

(55)

Step 3 Choose the third Lyapunov function

1198813(119905) =

1

2[1199042

3+

1

1205833

(1205883minus 1205883)2

+1

1205743

(1205793minus 1205793)2

] (56)

Taking time derivative gives

3(119905) = 119904

31199043+

1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

= 1199043(11986311990231198903+ 12058231198903) +

1

1205833

(1205883minus 1205883) 1205883

+1

1205743

(1205793minus 1205793)

1205793

(57)

Substituting the third state equation of (17) into (57) one has

3(119905) = 119904

3[minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3]

+ 1199043[Δ1198913+ 1198893+ 1199063+ 12058231198903]

+1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

(58)

We choose the third control law and adaptive law

1199063= 11989011198902minus 11989011199102minus 11989021199101+

(8 + 120572) 1198903

3minus 12058231198903

minus (1205883+ 1205793+ 1198963) sgn (119904

3)

(59)

1205883= 1205833

100381610038161003816100381611990431003816100381610038161003816

1205793= 1205743

100381610038161003816100381611990431003816100381610038161003816

(60)

Then the resulting derivative of 1198813is

3(119905) lt minus119896

3

100381610038161003816100381611990431003816100381610038161003816

(61)

6 Abstract and Applied Analysis

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus30

minus20

minus10

0

10

20

Time (s)

x(t)

(b)

0 50 100minus40

minus20

minus10

0

10

20

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 1 Chaotic trajectories with 120572 = 05

Step 4 Finally we gather the above three control functions as

119881 (119905) =1

2

3

sum

119894=1

1199042

119894+

3

sum

119894=1

1

2120583119894

(120588119894minus 120588119894)2

+

3

sum

119894=1

1

2120574119894

(120579119894minus 120579119894)2

(62)

It is clear from (48) (55) and (61) that

(119905) = minus (1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(63)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (64)

The resulting derivative of is

(1199041 1199042 1199043) lt minus119896 119904 (65)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (66)

By using Lemma 8 this Lyapunov function provides theproof of globally asymptotical stability with the control laws(46) (52) and (59) and adaptive laws (47) (53) and (60)

5 Numerical Simulations

51 Chaotic Behaviors of Fractional-Order Chaotic System In[55] the authors have provided us with numerical methodsof fractional calculus In [56] the chaotic behaviors of thefractional-order unified system were numerically investi-gated where it is found that the lowest order to exhibit chaosis 276

The chaotic behaviors are presented in Figures 1 and2 with fractional orders of 119902

1= 093 119902

2= 094 and

Abstract and Applied Analysis 7

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus40

minus20

0

20

40

Time (s)

x(t)

(b)

0 50 100minus40

minus20

0

20

40

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 2 Chaotic trajectories with 120572 = 1

1199023= 095 and the initial conditions of (119909

1(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1)Thenumerical algorithm is based on the followingGrunwald-Letnikovrsquos definition

119909 (119905119896) = (25120572 + 10) (119909

2(119905119896minus1

) minus 1199091(119905119896minus1

)) ℎ1199021

minus

119896

sum

119895=2

119888(1199021)

1198951199091(119905119896minus119895

)

1199092(119905119896) = [(28 minus 35120572) 119909

1(119905119896) minus 1199091(119905119896) 1199093(119905119896minus1

)

+ (29120572 minus 1) 1199092(119905119896minus1

)] ℎ1199022 minus

119896

sum

119895=2

119888(1199022)

1198951199092(119905119896minus119895

)

1199093(119905119896) = [119909

1(119905119896) 1199092(119905119896) minus

120572 + 8

31199093(119905119896minus1

)] ℎ1199023

minus

119896

sum

119895=2

119888(1199023)

1198951199093(119905119896minus119895

)

(67)

where 119879sim is the simulation time 119896 = 1 2 119873 for 119873 =

[119879simℎ]

52 Simulations of Synchronization of the Nominal SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed sliding modelcontrol in Section 41 In the numerical simulations thefractional orders are chosen as 119902

1= 093 119902

2= 094 and

1199023= 095 The initial conditions of the drive system (15) and

the response system (16) are chosen as (1199091(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1) and (1199101(0) 1199102(0) 1199103(0)) = (4 minus4 4) respectively

Parameters in (18) are chosen as 1205821= 1205822= 1205823= 4 Gains

of the control inputs in (24) (30) and (35) are chosen as1198961= 1198962= 1198962= 1

When 120572 = 05 and 120572 = 1 numerical simulations ofsynchronization of system (15) are presented in Figures 34 5 6 7 and 8 with control inputs (24) (30) and (35)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

8 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 3 Synchronization of determined fractional-order unified chaotic system with 120572 = 05 (blue line represents the trajectories of thedrive system while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 4 Synchronization errors states with 120572 = 05 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

0 2 4 6 8 10

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 5 Sliding surfaces states with 120572 = 05 (red line represents the first sliding surface state 1199041 blue line represents the second sliding

surface state 1199042 and black line represents the third sliding surface state 119904

3)

Abstract and Applied Analysis 9

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 6 Synchronization of determined fractional-order unified chaotic system with 120572 = 1 (blue line represents the trajectories of the drivesystem while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus2

minus1

Figure 7 Synchronization errors states with 120572 = 1 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

2 4 6 8 10

0

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 8 Sliding surfaces states with 120572 = 1 (red line represents the first sliding surface state 1199041 blue line represents the second sliding surface

state 1199042 and black line represents the third sliding surface state 119904

3)

10 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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

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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

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

Abstract and Applied Analysis 3

Lemma8 (see [53]) Assume that there exists a scalar function119881 of the state 119909with continuous first-order derivative such thatthe following are given

(i) 119881(119909) is positive definite(ii) (119909) is negative definite(iii) 119881(119909) rarr infin as 119909 rarr infin

Then the equilibrium at the origin is globally asymptoticallystable

3 Problem Formulation

In [54] Lu et al have considered a kind of chaotic systems andpointed out that these systems can be described in a unifiedform as follows

1= (25120572 + 10) (119909

2minus 1199091)

2= (28 minus 35120572) 119909

1minus 11990911199093+ (29120572 minus 1) 119909

2

3= 11990911199092minus

(8 + 120572) 1199093

3

(14)

where 1199091 1199092 and 119909

3are state variables and 120572 isin [0 1] is

the system parameter Lu et al [54] call system (14) a unifiedchaotic system because it is chaotic for any 120572 isin [0 1] When120572 isin [0 08) system (14) is called a the generalized Lorenzchaotic system When 120572 = 08 it is called the Lu chaoticsystem And it is called the generalized Chen chaotic systemwhen 120572 isin (08 1]

The fractional-order unified chaotic system has beenfirstly introduced and studied in [55] and reads as

11986311990211199091= (25120572 + 10) (119909

2minus 1199091)

11986311990221199092= (28 minus 35120572) 119909

1minus 11990911199093+ (29120572 minus 1) 119909

2

11986311990231199093= 11990911199092minus

(8 + 120572) 1199093

3

(15)

where 1199021 1199022 1199023isin (0 1] is the fractional order

System (15) is considered as the drive (master) system andthe response (slave) system is a controlled system as follows

11986311990211199101= (25120572 + 10) (119910

2minus 1199101) + 1199061

11986311990221199102= (28 minus 35120572) 119910

1minus 11991011199103+ (29120572 minus 1) 119910

2+ 1199062

11986311990231199103= 11991011199102minus

(8 + 120572) 1199103

3+ 1199063

(16)

Let us define the state errors between the response system(16) and the drive system (15) as 119890

1= 1199101minus 1199091 1198902= 1199102minus 1199092

and 1198903= 1199103minus 1199093

By subtracting (15) from (16) one can get the followingerror dynamical system

11986311990211198901= (25120572 + 10) (119890

2minus 1198901) + 1199061

11986311990221198902= (28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103

minus 11989031199101+ (29120572 minus 1) 119890

2+ 1199062

11986311990231198903= minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ 1199063

(17)

Ourmain objective in this paper is to investigate the synchro-nization issue for the fractional-order unified chaotic system(15) It is clear that the synchronization of systems (15) and(16) is equivalent to the stabilization of the error dynamicalsystem (17)

4 Synchronization Design

In the following the sliding mode control technique whichcan maintain low sensitivity to unmodeled dynamics andexternal disturbances is applied to establish an effectivecontrol law to guarantee the synchronization of the drivesystem (15) and the response system (16) Two major stepsare involved in the sliding mode control design firstlyconstructing an appropriate sliding surface on which thedesired system dynamics is stable and secondly developing asuitable control law such that the sliding condition is attained

41 Synchronization of the Nominal System In this sub-section let us firstly consider a simple case the nominalfractional-order unified chaotic system that is the systemcontains no systematic uncertainties or external disturbancesThe design procedure is elaborated in the rest part of thissubsection

411 Sliding Surfaces Design In order to achieve the stabilityof system (17) three sliding surfaces 119878

1 1198782 and 119878

3are

introduced as

119904119894(119905) = (119863

119902119894 + 120582119894) int

119905

0

119890119894(120591) 119889120591 119894 = 1 2 3 (18)

the time derivative of which becomes

119904119894(119905) = 119863

119902119894119890119894(119905) + 120582

119894119890119894(119905) 119894 = 1 2 3 (19)

As long as system (17) operates on the sliding surfacesit satisfies 119904

119894= 0 and 119904

119894= 0 119894 = 1 2 3 which yields the

following sliding mode dynamics

11986311990211198901= minus12058211198901

11986311990221198902= minus12058221198902

11986311990231198903= minus12058231198903

(20)

By using Lemmas 6 or 7 system (20) is asymptoticallystable As a result the sliding mode surfaces (18) we have justconstructed are appropriate for the control design

412 Control Laws Design

Step 1 Choose the first control Lyapunov function

1198811(119905) =

1

21199042

1 (21)

Taking time derivative gives

1(119905) = 119904

11199041= 1199041(11986311990211198901+ 12058211198901) (22)

4 Abstract and Applied Analysis

Substituting the first state equation of (17) into (22) one has

1(119905) = 119904

1[(25120572 + 10) (119890

2minus 1198901) + 1199061+ 12058211198901] (23)

Therefore by designing the first control law as

1199061= minus (25120572 + 10) (119890

2minus 1198901) minus 12058211198901minus 1198961sgn (119904

1) (24)

where 1198961gt 0 and

sgn (119904) =

1 119904 gt 0

0 119904 = 0

minus1 119904 lt 0

(25)

then (23) becomes

1(119905) = minus119896

1

100381610038161003816100381611990411003816100381610038161003816

(26)

Step 2 Choose the second control Lyapunov function

1198812(119905) =

1

21199042

2 (27)

By taking its derivative with respect to time yields

2(119905) = 119904

21199042= 1199042(11986311990221198902+ 12058221198902) (28)

Substituting the second state equation of (17) into (28) onehas

2(119905) = 119904

2[(28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103minus 11989031199101

+ (29120572 minus 1) 1198902+ 1199062+ 12058221198902]

(29)

Therefore by designing the second control law as

1199062= minus (28 minus 35120572) 119890

1minus 11989011198903+ 11989011199103+ 11989031199101

minus (29120572 minus 1) 1198902minus 12058221198902minus 1198962sgn (119904

3)

(30)

where 1198962gt 0

Equation (29) becomes

2(119905) = minus119896

2

100381610038161003816100381611990421003816100381610038161003816

(31)

Step 3 Choose the third control Lyapunov function

1198813(119905) =

1

21199042

3 (32)

Its time derivative is given by

3(119905) = 119904

31199043= 1199043(11986311990231198903+ 12058231198903) (33)

Substituting the third state equation of (17) into (33) one has

3(119905) = 119904

3[minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ 1199063+ 12058231198903]

(34)

We are then in the position to design the third control lawas follows

1199063= 11989011198902minus 11989011199102minus 11989021199101+

(8 + 120572) 1198903

3minus 12058231198903minus 1198963sgn (119904

3)

(35)

where 1198963gt 0

With this choice (34) can be rewritten as

3(119905) = minus119896

3

100381610038161003816100381611990431003816100381610038161003816

(36)

Step 4 Finally we gather the above three control functions as

119881 (1199041 1199042 1199043) =

1

21199042

1+

1

21199042

2+

1

21199042

3 (37)

It is clear from (26) (31) and (36) that

(1199041 1199042 1199043) = minus (119896

1

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(38)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (39)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (40)

The resulting derivative of 119881 is

(1199041 1199042 1199043) lt minus119896 119904 (41)

In terms of Lemma 8 the Lyapunov function (37) pro-vides the proof of globally asymptotical stability with thecontrol laws (24) (30) and (35)

42 Synchronization of the Uncertain System In this subsec-tion we will proceed to study the synchronization of thefractional-order unified chaotic system in the presence ofsystematic uncertainties and external disturbances which canbe hardly ignored in the real-world application It is assumedthat systematic uncertainties Δ119891

1 Δ1198912 and Δ119891

3and external

disturbances 1198891 1198892 and 119889

3are all bounded that is |Δ119891

119894| lt

120588119894and |119889

119894(119905)| lt 120579

119894 where 120588

119894and 120579

119894are unknown positive

constants 119894 = 1 2 3 Let us denote that 120588119894is an estimate of 120588

119894

while 120579119894is an estimate of 120579

119894 Since 120588

119894and 120579119894are unknown our

task in this subsection is fulfilled with an adaptive controllerconsisting of control laws and update laws to obtain 120588

119894and 120579119894

119894 = 1 2 3The uncertain fractional-order unified chaotic system can

be described as

11986311990211198901= (25120572 + 10) (119890

2minus 1198901) + Δ119891

1+ 1198891+ 1199061

11986311990221198902= (28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103minus 11989031199101

+ (29120572 minus 1) 1198902+ Δ1198912+ 1198892+ 1199062

11986311990231198903= minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ Δ1198913+ 1198893+ 1199063

(42)

The adaptive sliding mode design of system (42) consistsof four steps which are elaborated as follows

Step 1 Consider the first control Lyapunov function

1198811(119905) =

1

2[1199042

1+

1

1205831

(1205881minus 1205881)2

+1

1205741

(1205791minus 1205791)2

] (43)

Abstract and Applied Analysis 5

the time derivative of which becomes

1(119905) = 119904

11199041+

1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

= 1199041(11986311990211198901+ 12058211198901) +

1

1205831

(1205881minus 1205881) 1205881

+1

1205741

(1205791minus 1205791)

1205791

(44)

Substituting the first state equation of (17) into (44) one has

1(119905) = 119904

1[(25120572 + 10) (119890

2minus 1198901) + Δ119891

1+ 1198891+ 1199061+ 12058211198901]

+1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

(45)

By designing the first control law and adaptive law as

1199061= minus (25120572 + 10) (119890

2minus 1198901) minus 12058211198901minus (1205881+ 1205791+ 1198961) sgn (119904

1)

(46)

1205881= 1205831

100381610038161003816100381611990411003816100381610038161003816

1205791= 1205741

100381610038161003816100381611990411003816100381610038161003816

(47)

then (45) becomes

1= 119904 [(Δ119891

1+ 1198891) minus (120588

1+ 1205791+ 1198961) sgn (119904

1)]

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

le (1003816100381610038161003816Δ1198911

1003816100381610038161003816 +10038161003816100381610038161198891

1003816100381610038161003816)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

lt (1205881+ 1205791)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816 = minus1198961

100381610038161003816100381611990411003816100381610038161003816

(48)

Step 2 Choose the second control Lyapunov function

1198812(119905) =

1

2[1199042

2+

1

1205832

(1205882minus 1205882)2

+1

1205742

(1205792minus 1205792)2

] (49)

whose derivative is

2(119905) = 119904

21199042+

1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

= 1199042(11986311990221198902+ 12058221198902) +

1

1205832

(1205882minus 1205882) 1205882

+1

1205741

(1205792minus 1205792)

1205792

(50)

Substituting the second state equation of (17) into (50) onehas

2(119905) = 119904

2[(28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103

minus11989031199101+ (29120572 minus 1) 119890

2]

+ 1199042[Δ1198912+ 1198892+ 1199062+ 12058221198902]

+1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

(51)

We choose the control law and the adaptive law

1199062= minus (28 minus 35120572) 119890

1minus 11989011198903+ 11989011199103+ 11989031199101

minus (29120572 minus 1) 1198902minus 12058221198902minus (1205882+ 1205792+ 1198962) sgn (119904

2)

(52)

1205882= 1205832

100381610038161003816100381611990421003816100381610038161003816

(53)

1205792= 1205742

100381610038161003816100381611990421003816100381610038161003816

(54)

With this choice (51) becomes

2(119905) lt minus119896

2

100381610038161003816100381611990421003816100381610038161003816

(55)

Step 3 Choose the third Lyapunov function

1198813(119905) =

1

2[1199042

3+

1

1205833

(1205883minus 1205883)2

+1

1205743

(1205793minus 1205793)2

] (56)

Taking time derivative gives

3(119905) = 119904

31199043+

1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

= 1199043(11986311990231198903+ 12058231198903) +

1

1205833

(1205883minus 1205883) 1205883

+1

1205743

(1205793minus 1205793)

1205793

(57)

Substituting the third state equation of (17) into (57) one has

3(119905) = 119904

3[minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3]

+ 1199043[Δ1198913+ 1198893+ 1199063+ 12058231198903]

+1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

(58)

We choose the third control law and adaptive law

1199063= 11989011198902minus 11989011199102minus 11989021199101+

(8 + 120572) 1198903

3minus 12058231198903

minus (1205883+ 1205793+ 1198963) sgn (119904

3)

(59)

1205883= 1205833

100381610038161003816100381611990431003816100381610038161003816

1205793= 1205743

100381610038161003816100381611990431003816100381610038161003816

(60)

Then the resulting derivative of 1198813is

3(119905) lt minus119896

3

100381610038161003816100381611990431003816100381610038161003816

(61)

6 Abstract and Applied Analysis

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus30

minus20

minus10

0

10

20

Time (s)

x(t)

(b)

0 50 100minus40

minus20

minus10

0

10

20

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 1 Chaotic trajectories with 120572 = 05

Step 4 Finally we gather the above three control functions as

119881 (119905) =1

2

3

sum

119894=1

1199042

119894+

3

sum

119894=1

1

2120583119894

(120588119894minus 120588119894)2

+

3

sum

119894=1

1

2120574119894

(120579119894minus 120579119894)2

(62)

It is clear from (48) (55) and (61) that

(119905) = minus (1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(63)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (64)

The resulting derivative of is

(1199041 1199042 1199043) lt minus119896 119904 (65)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (66)

By using Lemma 8 this Lyapunov function provides theproof of globally asymptotical stability with the control laws(46) (52) and (59) and adaptive laws (47) (53) and (60)

5 Numerical Simulations

51 Chaotic Behaviors of Fractional-Order Chaotic System In[55] the authors have provided us with numerical methodsof fractional calculus In [56] the chaotic behaviors of thefractional-order unified system were numerically investi-gated where it is found that the lowest order to exhibit chaosis 276

The chaotic behaviors are presented in Figures 1 and2 with fractional orders of 119902

1= 093 119902

2= 094 and

Abstract and Applied Analysis 7

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus40

minus20

0

20

40

Time (s)

x(t)

(b)

0 50 100minus40

minus20

0

20

40

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 2 Chaotic trajectories with 120572 = 1

1199023= 095 and the initial conditions of (119909

1(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1)Thenumerical algorithm is based on the followingGrunwald-Letnikovrsquos definition

119909 (119905119896) = (25120572 + 10) (119909

2(119905119896minus1

) minus 1199091(119905119896minus1

)) ℎ1199021

minus

119896

sum

119895=2

119888(1199021)

1198951199091(119905119896minus119895

)

1199092(119905119896) = [(28 minus 35120572) 119909

1(119905119896) minus 1199091(119905119896) 1199093(119905119896minus1

)

+ (29120572 minus 1) 1199092(119905119896minus1

)] ℎ1199022 minus

119896

sum

119895=2

119888(1199022)

1198951199092(119905119896minus119895

)

1199093(119905119896) = [119909

1(119905119896) 1199092(119905119896) minus

120572 + 8

31199093(119905119896minus1

)] ℎ1199023

minus

119896

sum

119895=2

119888(1199023)

1198951199093(119905119896minus119895

)

(67)

where 119879sim is the simulation time 119896 = 1 2 119873 for 119873 =

[119879simℎ]

52 Simulations of Synchronization of the Nominal SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed sliding modelcontrol in Section 41 In the numerical simulations thefractional orders are chosen as 119902

1= 093 119902

2= 094 and

1199023= 095 The initial conditions of the drive system (15) and

the response system (16) are chosen as (1199091(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1) and (1199101(0) 1199102(0) 1199103(0)) = (4 minus4 4) respectively

Parameters in (18) are chosen as 1205821= 1205822= 1205823= 4 Gains

of the control inputs in (24) (30) and (35) are chosen as1198961= 1198962= 1198962= 1

When 120572 = 05 and 120572 = 1 numerical simulations ofsynchronization of system (15) are presented in Figures 34 5 6 7 and 8 with control inputs (24) (30) and (35)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

8 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 3 Synchronization of determined fractional-order unified chaotic system with 120572 = 05 (blue line represents the trajectories of thedrive system while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 4 Synchronization errors states with 120572 = 05 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

0 2 4 6 8 10

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 5 Sliding surfaces states with 120572 = 05 (red line represents the first sliding surface state 1199041 blue line represents the second sliding

surface state 1199042 and black line represents the third sliding surface state 119904

3)

Abstract and Applied Analysis 9

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 6 Synchronization of determined fractional-order unified chaotic system with 120572 = 1 (blue line represents the trajectories of the drivesystem while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus2

minus1

Figure 7 Synchronization errors states with 120572 = 1 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

2 4 6 8 10

0

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 8 Sliding surfaces states with 120572 = 1 (red line represents the first sliding surface state 1199041 blue line represents the second sliding surface

state 1199042 and black line represents the third sliding surface state 119904

3)

10 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Substituting the first state equation of (17) into (22) one has

1(119905) = 119904

1[(25120572 + 10) (119890

2minus 1198901) + 1199061+ 12058211198901] (23)

Therefore by designing the first control law as

1199061= minus (25120572 + 10) (119890

2minus 1198901) minus 12058211198901minus 1198961sgn (119904

1) (24)

where 1198961gt 0 and

sgn (119904) =

1 119904 gt 0

0 119904 = 0

minus1 119904 lt 0

(25)

then (23) becomes

1(119905) = minus119896

1

100381610038161003816100381611990411003816100381610038161003816

(26)

Step 2 Choose the second control Lyapunov function

1198812(119905) =

1

21199042

2 (27)

By taking its derivative with respect to time yields

2(119905) = 119904

21199042= 1199042(11986311990221198902+ 12058221198902) (28)

Substituting the second state equation of (17) into (28) onehas

2(119905) = 119904

2[(28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103minus 11989031199101

+ (29120572 minus 1) 1198902+ 1199062+ 12058221198902]

(29)

Therefore by designing the second control law as

1199062= minus (28 minus 35120572) 119890

1minus 11989011198903+ 11989011199103+ 11989031199101

minus (29120572 minus 1) 1198902minus 12058221198902minus 1198962sgn (119904

3)

(30)

where 1198962gt 0

Equation (29) becomes

2(119905) = minus119896

2

100381610038161003816100381611990421003816100381610038161003816

(31)

Step 3 Choose the third control Lyapunov function

1198813(119905) =

1

21199042

3 (32)

Its time derivative is given by

3(119905) = 119904

31199043= 1199043(11986311990231198903+ 12058231198903) (33)

Substituting the third state equation of (17) into (33) one has

3(119905) = 119904

3[minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ 1199063+ 12058231198903]

(34)

We are then in the position to design the third control lawas follows

1199063= 11989011198902minus 11989011199102minus 11989021199101+

(8 + 120572) 1198903

3minus 12058231198903minus 1198963sgn (119904

3)

(35)

where 1198963gt 0

With this choice (34) can be rewritten as

3(119905) = minus119896

3

100381610038161003816100381611990431003816100381610038161003816

(36)

Step 4 Finally we gather the above three control functions as

119881 (1199041 1199042 1199043) =

1

21199042

1+

1

21199042

2+

1

21199042

3 (37)

It is clear from (26) (31) and (36) that

(1199041 1199042 1199043) = minus (119896

1

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(38)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (39)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (40)

The resulting derivative of 119881 is

(1199041 1199042 1199043) lt minus119896 119904 (41)

In terms of Lemma 8 the Lyapunov function (37) pro-vides the proof of globally asymptotical stability with thecontrol laws (24) (30) and (35)

42 Synchronization of the Uncertain System In this subsec-tion we will proceed to study the synchronization of thefractional-order unified chaotic system in the presence ofsystematic uncertainties and external disturbances which canbe hardly ignored in the real-world application It is assumedthat systematic uncertainties Δ119891

1 Δ1198912 and Δ119891

3and external

disturbances 1198891 1198892 and 119889

3are all bounded that is |Δ119891

119894| lt

120588119894and |119889

119894(119905)| lt 120579

119894 where 120588

119894and 120579

119894are unknown positive

constants 119894 = 1 2 3 Let us denote that 120588119894is an estimate of 120588

119894

while 120579119894is an estimate of 120579

119894 Since 120588

119894and 120579119894are unknown our

task in this subsection is fulfilled with an adaptive controllerconsisting of control laws and update laws to obtain 120588

119894and 120579119894

119894 = 1 2 3The uncertain fractional-order unified chaotic system can

be described as

11986311990211198901= (25120572 + 10) (119890

2minus 1198901) + Δ119891

1+ 1198891+ 1199061

11986311990221198902= (28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103minus 11989031199101

+ (29120572 minus 1) 1198902+ Δ1198912+ 1198892+ 1199062

11986311990231198903= minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3+ Δ1198913+ 1198893+ 1199063

(42)

The adaptive sliding mode design of system (42) consistsof four steps which are elaborated as follows

Step 1 Consider the first control Lyapunov function

1198811(119905) =

1

2[1199042

1+

1

1205831

(1205881minus 1205881)2

+1

1205741

(1205791minus 1205791)2

] (43)

Abstract and Applied Analysis 5

the time derivative of which becomes

1(119905) = 119904

11199041+

1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

= 1199041(11986311990211198901+ 12058211198901) +

1

1205831

(1205881minus 1205881) 1205881

+1

1205741

(1205791minus 1205791)

1205791

(44)

Substituting the first state equation of (17) into (44) one has

1(119905) = 119904

1[(25120572 + 10) (119890

2minus 1198901) + Δ119891

1+ 1198891+ 1199061+ 12058211198901]

+1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

(45)

By designing the first control law and adaptive law as

1199061= minus (25120572 + 10) (119890

2minus 1198901) minus 12058211198901minus (1205881+ 1205791+ 1198961) sgn (119904

1)

(46)

1205881= 1205831

100381610038161003816100381611990411003816100381610038161003816

1205791= 1205741

100381610038161003816100381611990411003816100381610038161003816

(47)

then (45) becomes

1= 119904 [(Δ119891

1+ 1198891) minus (120588

1+ 1205791+ 1198961) sgn (119904

1)]

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

le (1003816100381610038161003816Δ1198911

1003816100381610038161003816 +10038161003816100381610038161198891

1003816100381610038161003816)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

lt (1205881+ 1205791)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816 = minus1198961

100381610038161003816100381611990411003816100381610038161003816

(48)

Step 2 Choose the second control Lyapunov function

1198812(119905) =

1

2[1199042

2+

1

1205832

(1205882minus 1205882)2

+1

1205742

(1205792minus 1205792)2

] (49)

whose derivative is

2(119905) = 119904

21199042+

1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

= 1199042(11986311990221198902+ 12058221198902) +

1

1205832

(1205882minus 1205882) 1205882

+1

1205741

(1205792minus 1205792)

1205792

(50)

Substituting the second state equation of (17) into (50) onehas

2(119905) = 119904

2[(28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103

minus11989031199101+ (29120572 minus 1) 119890

2]

+ 1199042[Δ1198912+ 1198892+ 1199062+ 12058221198902]

+1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

(51)

We choose the control law and the adaptive law

1199062= minus (28 minus 35120572) 119890

1minus 11989011198903+ 11989011199103+ 11989031199101

minus (29120572 minus 1) 1198902minus 12058221198902minus (1205882+ 1205792+ 1198962) sgn (119904

2)

(52)

1205882= 1205832

100381610038161003816100381611990421003816100381610038161003816

(53)

1205792= 1205742

100381610038161003816100381611990421003816100381610038161003816

(54)

With this choice (51) becomes

2(119905) lt minus119896

2

100381610038161003816100381611990421003816100381610038161003816

(55)

Step 3 Choose the third Lyapunov function

1198813(119905) =

1

2[1199042

3+

1

1205833

(1205883minus 1205883)2

+1

1205743

(1205793minus 1205793)2

] (56)

Taking time derivative gives

3(119905) = 119904

31199043+

1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

= 1199043(11986311990231198903+ 12058231198903) +

1

1205833

(1205883minus 1205883) 1205883

+1

1205743

(1205793minus 1205793)

1205793

(57)

Substituting the third state equation of (17) into (57) one has

3(119905) = 119904

3[minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3]

+ 1199043[Δ1198913+ 1198893+ 1199063+ 12058231198903]

+1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

(58)

We choose the third control law and adaptive law

1199063= 11989011198902minus 11989011199102minus 11989021199101+

(8 + 120572) 1198903

3minus 12058231198903

minus (1205883+ 1205793+ 1198963) sgn (119904

3)

(59)

1205883= 1205833

100381610038161003816100381611990431003816100381610038161003816

1205793= 1205743

100381610038161003816100381611990431003816100381610038161003816

(60)

Then the resulting derivative of 1198813is

3(119905) lt minus119896

3

100381610038161003816100381611990431003816100381610038161003816

(61)

6 Abstract and Applied Analysis

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus30

minus20

minus10

0

10

20

Time (s)

x(t)

(b)

0 50 100minus40

minus20

minus10

0

10

20

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 1 Chaotic trajectories with 120572 = 05

Step 4 Finally we gather the above three control functions as

119881 (119905) =1

2

3

sum

119894=1

1199042

119894+

3

sum

119894=1

1

2120583119894

(120588119894minus 120588119894)2

+

3

sum

119894=1

1

2120574119894

(120579119894minus 120579119894)2

(62)

It is clear from (48) (55) and (61) that

(119905) = minus (1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(63)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (64)

The resulting derivative of is

(1199041 1199042 1199043) lt minus119896 119904 (65)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (66)

By using Lemma 8 this Lyapunov function provides theproof of globally asymptotical stability with the control laws(46) (52) and (59) and adaptive laws (47) (53) and (60)

5 Numerical Simulations

51 Chaotic Behaviors of Fractional-Order Chaotic System In[55] the authors have provided us with numerical methodsof fractional calculus In [56] the chaotic behaviors of thefractional-order unified system were numerically investi-gated where it is found that the lowest order to exhibit chaosis 276

The chaotic behaviors are presented in Figures 1 and2 with fractional orders of 119902

1= 093 119902

2= 094 and

Abstract and Applied Analysis 7

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus40

minus20

0

20

40

Time (s)

x(t)

(b)

0 50 100minus40

minus20

0

20

40

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 2 Chaotic trajectories with 120572 = 1

1199023= 095 and the initial conditions of (119909

1(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1)Thenumerical algorithm is based on the followingGrunwald-Letnikovrsquos definition

119909 (119905119896) = (25120572 + 10) (119909

2(119905119896minus1

) minus 1199091(119905119896minus1

)) ℎ1199021

minus

119896

sum

119895=2

119888(1199021)

1198951199091(119905119896minus119895

)

1199092(119905119896) = [(28 minus 35120572) 119909

1(119905119896) minus 1199091(119905119896) 1199093(119905119896minus1

)

+ (29120572 minus 1) 1199092(119905119896minus1

)] ℎ1199022 minus

119896

sum

119895=2

119888(1199022)

1198951199092(119905119896minus119895

)

1199093(119905119896) = [119909

1(119905119896) 1199092(119905119896) minus

120572 + 8

31199093(119905119896minus1

)] ℎ1199023

minus

119896

sum

119895=2

119888(1199023)

1198951199093(119905119896minus119895

)

(67)

where 119879sim is the simulation time 119896 = 1 2 119873 for 119873 =

[119879simℎ]

52 Simulations of Synchronization of the Nominal SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed sliding modelcontrol in Section 41 In the numerical simulations thefractional orders are chosen as 119902

1= 093 119902

2= 094 and

1199023= 095 The initial conditions of the drive system (15) and

the response system (16) are chosen as (1199091(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1) and (1199101(0) 1199102(0) 1199103(0)) = (4 minus4 4) respectively

Parameters in (18) are chosen as 1205821= 1205822= 1205823= 4 Gains

of the control inputs in (24) (30) and (35) are chosen as1198961= 1198962= 1198962= 1

When 120572 = 05 and 120572 = 1 numerical simulations ofsynchronization of system (15) are presented in Figures 34 5 6 7 and 8 with control inputs (24) (30) and (35)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

8 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 3 Synchronization of determined fractional-order unified chaotic system with 120572 = 05 (blue line represents the trajectories of thedrive system while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 4 Synchronization errors states with 120572 = 05 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

0 2 4 6 8 10

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 5 Sliding surfaces states with 120572 = 05 (red line represents the first sliding surface state 1199041 blue line represents the second sliding

surface state 1199042 and black line represents the third sliding surface state 119904

3)

Abstract and Applied Analysis 9

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 6 Synchronization of determined fractional-order unified chaotic system with 120572 = 1 (blue line represents the trajectories of the drivesystem while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus2

minus1

Figure 7 Synchronization errors states with 120572 = 1 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

2 4 6 8 10

0

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 8 Sliding surfaces states with 120572 = 1 (red line represents the first sliding surface state 1199041 blue line represents the second sliding surface

state 1199042 and black line represents the third sliding surface state 119904

3)

10 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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

Abstract and Applied Analysis 5

the time derivative of which becomes

1(119905) = 119904

11199041+

1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

= 1199041(11986311990211198901+ 12058211198901) +

1

1205831

(1205881minus 1205881) 1205881

+1

1205741

(1205791minus 1205791)

1205791

(44)

Substituting the first state equation of (17) into (44) one has

1(119905) = 119904

1[(25120572 + 10) (119890

2minus 1198901) + Δ119891

1+ 1198891+ 1199061+ 12058211198901]

+1

1205831

(1205881minus 1205881) 1205881+

1

1205741

(1205791minus 1205791)

1205791

(45)

By designing the first control law and adaptive law as

1199061= minus (25120572 + 10) (119890

2minus 1198901) minus 12058211198901minus (1205881+ 1205791+ 1198961) sgn (119904

1)

(46)

1205881= 1205831

100381610038161003816100381611990411003816100381610038161003816

1205791= 1205741

100381610038161003816100381611990411003816100381610038161003816

(47)

then (45) becomes

1= 119904 [(Δ119891

1+ 1198891) minus (120588

1+ 1205791+ 1198961) sgn (119904

1)]

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

le (1003816100381610038161003816Δ1198911

1003816100381610038161003816 +10038161003816100381610038161198891

1003816100381610038161003816)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816

lt (1205881+ 1205791)10038161003816100381610038161199041

1003816100381610038161003816 minus (1205881+ 1205791+ 1198961)10038161003816100381610038161199041

1003816100381610038161003816

+ (1205881minus 1205881)10038161003816100381610038161199041

1003816100381610038161003816 + (1205791minus 1205791)10038161003816100381610038161199041

1003816100381610038161003816 = minus1198961

100381610038161003816100381611990411003816100381610038161003816

(48)

Step 2 Choose the second control Lyapunov function

1198812(119905) =

1

2[1199042

2+

1

1205832

(1205882minus 1205882)2

+1

1205742

(1205792minus 1205792)2

] (49)

whose derivative is

2(119905) = 119904

21199042+

1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

= 1199042(11986311990221198902+ 12058221198902) +

1

1205832

(1205882minus 1205882) 1205882

+1

1205741

(1205792minus 1205792)

1205792

(50)

Substituting the second state equation of (17) into (50) onehas

2(119905) = 119904

2[(28 minus 35120572) 119890

1+ 11989011198903minus 11989011199103

minus11989031199101+ (29120572 minus 1) 119890

2]

+ 1199042[Δ1198912+ 1198892+ 1199062+ 12058221198902]

+1

1205832

(1205882minus 1205882) 1205882+

1

1205742

(1205792minus 1205792)

1205792

(51)

We choose the control law and the adaptive law

1199062= minus (28 minus 35120572) 119890

1minus 11989011198903+ 11989011199103+ 11989031199101

minus (29120572 minus 1) 1198902minus 12058221198902minus (1205882+ 1205792+ 1198962) sgn (119904

2)

(52)

1205882= 1205832

100381610038161003816100381611990421003816100381610038161003816

(53)

1205792= 1205742

100381610038161003816100381611990421003816100381610038161003816

(54)

With this choice (51) becomes

2(119905) lt minus119896

2

100381610038161003816100381611990421003816100381610038161003816

(55)

Step 3 Choose the third Lyapunov function

1198813(119905) =

1

2[1199042

3+

1

1205833

(1205883minus 1205883)2

+1

1205743

(1205793minus 1205793)2

] (56)

Taking time derivative gives

3(119905) = 119904

31199043+

1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

= 1199043(11986311990231198903+ 12058231198903) +

1

1205833

(1205883minus 1205883) 1205883

+1

1205743

(1205793minus 1205793)

1205793

(57)

Substituting the third state equation of (17) into (57) one has

3(119905) = 119904

3[minus11989011198902+ 11989011199102+ 11989021199101minus

(8 + 120572) 1198903

3]

+ 1199043[Δ1198913+ 1198893+ 1199063+ 12058231198903]

+1

1205833

(1205883minus 1205883) 1205883+

1

1205743

(1205793minus 1205793)

1205793

(58)

We choose the third control law and adaptive law

1199063= 11989011198902minus 11989011199102minus 11989021199101+

(8 + 120572) 1198903

3minus 12058231198903

minus (1205883+ 1205793+ 1198963) sgn (119904

3)

(59)

1205883= 1205833

100381610038161003816100381611990431003816100381610038161003816

1205793= 1205743

100381610038161003816100381611990431003816100381610038161003816

(60)

Then the resulting derivative of 1198813is

3(119905) lt minus119896

3

100381610038161003816100381611990431003816100381610038161003816

(61)

6 Abstract and Applied Analysis

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus30

minus20

minus10

0

10

20

Time (s)

x(t)

(b)

0 50 100minus40

minus20

minus10

0

10

20

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 1 Chaotic trajectories with 120572 = 05

Step 4 Finally we gather the above three control functions as

119881 (119905) =1

2

3

sum

119894=1

1199042

119894+

3

sum

119894=1

1

2120583119894

(120588119894minus 120588119894)2

+

3

sum

119894=1

1

2120574119894

(120579119894minus 120579119894)2

(62)

It is clear from (48) (55) and (61) that

(119905) = minus (1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(63)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (64)

The resulting derivative of is

(1199041 1199042 1199043) lt minus119896 119904 (65)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (66)

By using Lemma 8 this Lyapunov function provides theproof of globally asymptotical stability with the control laws(46) (52) and (59) and adaptive laws (47) (53) and (60)

5 Numerical Simulations

51 Chaotic Behaviors of Fractional-Order Chaotic System In[55] the authors have provided us with numerical methodsof fractional calculus In [56] the chaotic behaviors of thefractional-order unified system were numerically investi-gated where it is found that the lowest order to exhibit chaosis 276

The chaotic behaviors are presented in Figures 1 and2 with fractional orders of 119902

1= 093 119902

2= 094 and

Abstract and Applied Analysis 7

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus40

minus20

0

20

40

Time (s)

x(t)

(b)

0 50 100minus40

minus20

0

20

40

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 2 Chaotic trajectories with 120572 = 1

1199023= 095 and the initial conditions of (119909

1(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1)Thenumerical algorithm is based on the followingGrunwald-Letnikovrsquos definition

119909 (119905119896) = (25120572 + 10) (119909

2(119905119896minus1

) minus 1199091(119905119896minus1

)) ℎ1199021

minus

119896

sum

119895=2

119888(1199021)

1198951199091(119905119896minus119895

)

1199092(119905119896) = [(28 minus 35120572) 119909

1(119905119896) minus 1199091(119905119896) 1199093(119905119896minus1

)

+ (29120572 minus 1) 1199092(119905119896minus1

)] ℎ1199022 minus

119896

sum

119895=2

119888(1199022)

1198951199092(119905119896minus119895

)

1199093(119905119896) = [119909

1(119905119896) 1199092(119905119896) minus

120572 + 8

31199093(119905119896minus1

)] ℎ1199023

minus

119896

sum

119895=2

119888(1199023)

1198951199093(119905119896minus119895

)

(67)

where 119879sim is the simulation time 119896 = 1 2 119873 for 119873 =

[119879simℎ]

52 Simulations of Synchronization of the Nominal SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed sliding modelcontrol in Section 41 In the numerical simulations thefractional orders are chosen as 119902

1= 093 119902

2= 094 and

1199023= 095 The initial conditions of the drive system (15) and

the response system (16) are chosen as (1199091(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1) and (1199101(0) 1199102(0) 1199103(0)) = (4 minus4 4) respectively

Parameters in (18) are chosen as 1205821= 1205822= 1205823= 4 Gains

of the control inputs in (24) (30) and (35) are chosen as1198961= 1198962= 1198962= 1

When 120572 = 05 and 120572 = 1 numerical simulations ofsynchronization of system (15) are presented in Figures 34 5 6 7 and 8 with control inputs (24) (30) and (35)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

8 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 3 Synchronization of determined fractional-order unified chaotic system with 120572 = 05 (blue line represents the trajectories of thedrive system while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 4 Synchronization errors states with 120572 = 05 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

0 2 4 6 8 10

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 5 Sliding surfaces states with 120572 = 05 (red line represents the first sliding surface state 1199041 blue line represents the second sliding

surface state 1199042 and black line represents the third sliding surface state 119904

3)

Abstract and Applied Analysis 9

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 6 Synchronization of determined fractional-order unified chaotic system with 120572 = 1 (blue line represents the trajectories of the drivesystem while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus2

minus1

Figure 7 Synchronization errors states with 120572 = 1 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

2 4 6 8 10

0

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 8 Sliding surfaces states with 120572 = 1 (red line represents the first sliding surface state 1199041 blue line represents the second sliding surface

state 1199042 and black line represents the third sliding surface state 119904

3)

10 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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 Abstract and Applied Analysis

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus30

minus20

minus10

0

10

20

Time (s)

x(t)

(b)

0 50 100minus40

minus20

minus10

0

10

20

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 1 Chaotic trajectories with 120572 = 05

Step 4 Finally we gather the above three control functions as

119881 (119905) =1

2

3

sum

119894=1

1199042

119894+

3

sum

119894=1

1

2120583119894

(120588119894minus 120588119894)2

+

3

sum

119894=1

1

2120574119894

(120579119894minus 120579119894)2

(62)

It is clear from (48) (55) and (61) that

(119905) = minus (1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816)

(63)

There exists some 119896 gt 0 such that

1198961

100381610038161003816100381611990411003816100381610038161003816 + 1198962

100381610038161003816100381611990421003816100381610038161003816 + 1198963

100381610038161003816100381611990431003816100381610038161003816 gt 119896 119904 (64)

The resulting derivative of is

(1199041 1199042 1199043) lt minus119896 119904 (65)

where

119904 = radic1199042

1+ 1199042

2+ 1199042

3 (66)

By using Lemma 8 this Lyapunov function provides theproof of globally asymptotical stability with the control laws(46) (52) and (59) and adaptive laws (47) (53) and (60)

5 Numerical Simulations

51 Chaotic Behaviors of Fractional-Order Chaotic System In[55] the authors have provided us with numerical methodsof fractional calculus In [56] the chaotic behaviors of thefractional-order unified system were numerically investi-gated where it is found that the lowest order to exhibit chaosis 276

The chaotic behaviors are presented in Figures 1 and2 with fractional orders of 119902

1= 093 119902

2= 094 and

Abstract and Applied Analysis 7

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus40

minus20

0

20

40

Time (s)

x(t)

(b)

0 50 100minus40

minus20

0

20

40

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 2 Chaotic trajectories with 120572 = 1

1199023= 095 and the initial conditions of (119909

1(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1)Thenumerical algorithm is based on the followingGrunwald-Letnikovrsquos definition

119909 (119905119896) = (25120572 + 10) (119909

2(119905119896minus1

) minus 1199091(119905119896minus1

)) ℎ1199021

minus

119896

sum

119895=2

119888(1199021)

1198951199091(119905119896minus119895

)

1199092(119905119896) = [(28 minus 35120572) 119909

1(119905119896) minus 1199091(119905119896) 1199093(119905119896minus1

)

+ (29120572 minus 1) 1199092(119905119896minus1

)] ℎ1199022 minus

119896

sum

119895=2

119888(1199022)

1198951199092(119905119896minus119895

)

1199093(119905119896) = [119909

1(119905119896) 1199092(119905119896) minus

120572 + 8

31199093(119905119896minus1

)] ℎ1199023

minus

119896

sum

119895=2

119888(1199023)

1198951199093(119905119896minus119895

)

(67)

where 119879sim is the simulation time 119896 = 1 2 119873 for 119873 =

[119879simℎ]

52 Simulations of Synchronization of the Nominal SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed sliding modelcontrol in Section 41 In the numerical simulations thefractional orders are chosen as 119902

1= 093 119902

2= 094 and

1199023= 095 The initial conditions of the drive system (15) and

the response system (16) are chosen as (1199091(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1) and (1199101(0) 1199102(0) 1199103(0)) = (4 minus4 4) respectively

Parameters in (18) are chosen as 1205821= 1205822= 1205823= 4 Gains

of the control inputs in (24) (30) and (35) are chosen as1198961= 1198962= 1198962= 1

When 120572 = 05 and 120572 = 1 numerical simulations ofsynchronization of system (15) are presented in Figures 34 5 6 7 and 8 with control inputs (24) (30) and (35)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

8 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 3 Synchronization of determined fractional-order unified chaotic system with 120572 = 05 (blue line represents the trajectories of thedrive system while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 4 Synchronization errors states with 120572 = 05 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

0 2 4 6 8 10

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 5 Sliding surfaces states with 120572 = 05 (red line represents the first sliding surface state 1199041 blue line represents the second sliding

surface state 1199042 and black line represents the third sliding surface state 119904

3)

Abstract and Applied Analysis 9

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 6 Synchronization of determined fractional-order unified chaotic system with 120572 = 1 (blue line represents the trajectories of the drivesystem while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus2

minus1

Figure 7 Synchronization errors states with 120572 = 1 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

2 4 6 8 10

0

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 8 Sliding surfaces states with 120572 = 1 (red line represents the first sliding surface state 1199041 blue line represents the second sliding surface

state 1199042 and black line represents the third sliding surface state 119904

3)

10 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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

Abstract and Applied Analysis 7

0

500

500

50

z(t)

y(t)x(t

)

minus50minus50

(a)

0 50 100minus40

minus20

0

20

40

Time (s)

x(t)

(b)

0 50 100minus40

minus20

0

20

40

Time (s)

y(t)

(c)

50

40

30

20

10

00 50 100

Time (s)

z(t)

(d)

Figure 2 Chaotic trajectories with 120572 = 1

1199023= 095 and the initial conditions of (119909

1(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1)Thenumerical algorithm is based on the followingGrunwald-Letnikovrsquos definition

119909 (119905119896) = (25120572 + 10) (119909

2(119905119896minus1

) minus 1199091(119905119896minus1

)) ℎ1199021

minus

119896

sum

119895=2

119888(1199021)

1198951199091(119905119896minus119895

)

1199092(119905119896) = [(28 minus 35120572) 119909

1(119905119896) minus 1199091(119905119896) 1199093(119905119896minus1

)

+ (29120572 minus 1) 1199092(119905119896minus1

)] ℎ1199022 minus

119896

sum

119895=2

119888(1199022)

1198951199092(119905119896minus119895

)

1199093(119905119896) = [119909

1(119905119896) 1199092(119905119896) minus

120572 + 8

31199093(119905119896minus1

)] ℎ1199023

minus

119896

sum

119895=2

119888(1199023)

1198951199093(119905119896minus119895

)

(67)

where 119879sim is the simulation time 119896 = 1 2 119873 for 119873 =

[119879simℎ]

52 Simulations of Synchronization of the Nominal SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed sliding modelcontrol in Section 41 In the numerical simulations thefractional orders are chosen as 119902

1= 093 119902

2= 094 and

1199023= 095 The initial conditions of the drive system (15) and

the response system (16) are chosen as (1199091(0) 1199092(0) 1199093(0)) =

(minus1 minus2 1) and (1199101(0) 1199102(0) 1199103(0)) = (4 minus4 4) respectively

Parameters in (18) are chosen as 1205821= 1205822= 1205823= 4 Gains

of the control inputs in (24) (30) and (35) are chosen as1198961= 1198962= 1198962= 1

When 120572 = 05 and 120572 = 1 numerical simulations ofsynchronization of system (15) are presented in Figures 34 5 6 7 and 8 with control inputs (24) (30) and (35)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

8 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 3 Synchronization of determined fractional-order unified chaotic system with 120572 = 05 (blue line represents the trajectories of thedrive system while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 4 Synchronization errors states with 120572 = 05 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

0 2 4 6 8 10

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 5 Sliding surfaces states with 120572 = 05 (red line represents the first sliding surface state 1199041 blue line represents the second sliding

surface state 1199042 and black line represents the third sliding surface state 119904

3)

Abstract and Applied Analysis 9

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 6 Synchronization of determined fractional-order unified chaotic system with 120572 = 1 (blue line represents the trajectories of the drivesystem while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus2

minus1

Figure 7 Synchronization errors states with 120572 = 1 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

2 4 6 8 10

0

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 8 Sliding surfaces states with 120572 = 1 (red line represents the first sliding surface state 1199041 blue line represents the second sliding surface

state 1199042 and black line represents the third sliding surface state 119904

3)

10 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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

8 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 3 Synchronization of determined fractional-order unified chaotic system with 120572 = 05 (blue line represents the trajectories of thedrive system while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 4 Synchronization errors states with 120572 = 05 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

0 2 4 6 8 10

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 5 Sliding surfaces states with 120572 = 05 (red line represents the first sliding surface state 1199041 blue line represents the second sliding

surface state 1199042 and black line represents the third sliding surface state 119904

3)

Abstract and Applied Analysis 9

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 6 Synchronization of determined fractional-order unified chaotic system with 120572 = 1 (blue line represents the trajectories of the drivesystem while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus2

minus1

Figure 7 Synchronization errors states with 120572 = 1 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

2 4 6 8 10

0

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 8 Sliding surfaces states with 120572 = 1 (red line represents the first sliding surface state 1199041 blue line represents the second sliding surface

state 1199042 and black line represents the third sliding surface state 119904

3)

10 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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

Abstract and Applied Analysis 9

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 6 Synchronization of determined fractional-order unified chaotic system with 120572 = 1 (blue line represents the trajectories of the drivesystem while red line represents the trajectories of the response system)

0 2 4 6 8 10

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus2

minus1

Figure 7 Synchronization errors states with 120572 = 1 (red line represents the first error state 1198901 blue line represents the second error state 119890

2

and black line represents the third error state 1198903)

2 4 6 8 10

0

0

10

20

30

40

50

60

Time (s)

Slid

ing

surfa

ces

minus10

minus20

Figure 8 Sliding surfaces states with 120572 = 1 (red line represents the first sliding surface state 1199041 blue line represents the second sliding surface

state 1199042 and black line represents the third sliding surface state 119904

3)

10 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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 Abstract and Applied Analysis

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 9 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 05 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

0 1 2 3 4 5

0

1

2

3

4

5

Time (s)

Sync

hron

izat

ion

erro

rs

minus1

minus2

Figure 10 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 05 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

53 Simulations of Synchronization of the Uncertain SystemIn this subsection numerical simulations are presented todemonstrate the effectiveness of the proposed adaptive slid-ing model control in Section 42 In the numerical simula-tions the fractional orders are always chosen as 119902

1= 093

1199022= 094 and 119902

3= 095 The initial conditions of the drive

system (15) and the response system (16) are also chosen as(1199091(0) 1199092(0) 1199093(0)) = (minus1 minus2 1) and (119910

1(0) 1199102(0) 1199103(0)) =

(4 minus4 4) respectively Parameters in (18) are chosen as

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus10

minus5

Figure 11 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 05 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

0 2 4 6 8 10

0

50

Time (s)

minus50

x1y

1

(a)

0 2 4 6 8 10

0

50

Time (s)

minus50

x2y

2

(b)

0 2 4 6 8 100

50

Time (s)

x3y

3

(c)

Figure 12 Synchronization of fractional-order unified chaoticsystem in the presence of systematic uncertainties and externaldisturbances with 120572 = 1 (blue line represents the trajectories ofthe drive system while red line represents the trajectories of theresponse system)

1205821

= 1205822

= 1205823

= 20 Gains of the control laws (46) (52)and (59) are chosen as 119896

1= 1198962

= 1198962

= 05 Gains of theadaptive laws (47) (53) and (60) are chosen as 120583

1= 1205741=

1205832

= 1205742

= 1205833

= 1205743

= 01 Systematic uncertainties andexternal disturbances are assumed to be Δ119891

1= 0 119889

1=

minus3 cos(120587(119905 minus 01)) Δ1198912= minus06 sin(2(119910

1minus 1199091)) 1198892= 5 sin(119905)

Δ1198913= 0 and 119889

3= 2 cos(119905 minus 01)

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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

Abstract and Applied Analysis 11

0 1 2 3 4 5

0

1

2

3

4

5

Sync

hron

izat

ion

erro

rs

minus1

minus2

Time (s)

Figure 13 Synchronization errors states in the presence of system-atic uncertainties and external disturbances with 120572 = 1 (red linerepresents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 1 2 3 4 5

0

5

10

15

Time (s)

Slid

ing

surfa

ces

minus5

minus10

Figure 14 Sliding surfaces states in the presence of systematicuncertainties and external disturbances with 120572 = 1 (red linerepresents the first sliding surface state 119904

1 blue line represents the

second sliding surface state 1199042 and black line represents the third

sliding surface state 1199043)

When 120572 = 05 and 120572 = 1 numerical simulations arepresented in Figures 9 10 11 12 13 and 14 with control inputs(46) (52) and (59) and adaptive laws (47) (53) and (60)For interpretations of the references to colors in these figurelegends the reader is referred to the web version of this paper

6 Conclusions

This work is concerned with robust synchronization of thefractional-order unified chaotic system The sliding modecontrol technique was applied to propose the control designof nominal system and adaptive slidingmode control scheme

was designed to develop the control laws and adaptive laws foruncertain system with systematic uncertainties and externaldisturbances whose bounds are unknown Numerical simu-lations were presented to demonstrate the effectiveness of thetwo kinds of techniques

References

[1] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 Springer Berlin Germany 2010

[2] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[3] Y Chen and I Petras ldquoFractional order control a tutorialrdquo inProceedings of the American Control Conference Hyatt RegencyRiverfront St Louis MO USA 2009

[4] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[5] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[6] YHWang ldquoMittag-Leffler stability of a hybrid ratio-dependentfractional three species food chainrdquo Communications in Frac-tional Calculus vol 3 pp 111ndash118 2012

[7] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[8] I Petras ldquoA note on the fractional-order Chuarsquos systemrdquo ChaosSolitons and Fractals vol 38 no 1 pp 140ndash147 2008

[9] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[10] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[11] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[12] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 Article ID 034101 2003

[13] Y Yu H-X Li S Wang and J Yu ldquoDynamic analysis of afractional-order Lorenz chaotic systemrdquo Chaos Solitons andFractals vol 42 no 2 pp 1181ndash1189 2009

[14] C Li and G Chen ldquoChaos in the fractional order Chen systemand its controlrdquo Chaos Solitons and Fractals vol 22 no 3 pp549ndash554 2004

[15] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[16] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[17] J G Lu ldquoChaotic dynamics of the fractional-order Lu systemand its synchronizationrdquo Physics Letters A vol 354 no 4 pp305ndash311 2006

12 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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 Abstract and Applied Analysis

[18] X-Y Wang and M-J Wang ldquoDynamic analysis of thefractional-order Liu systemand its synchronizationrdquoChaos vol17 no 3 Article ID 033106 2007

[19] C Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[20] W Zhang S Zhou H Li and H Zhu ldquoChaos in a fractional-order Rossler systemrdquo Chaos Solitons and Fractals vol 42 no3 pp 1684ndash1691 2009

[21] J G Lu ldquoChaotic dynamics and synchronization of fractional-order Arneodorsquos systemsrdquo Chaos Solitons and Fractals vol 26no 4 pp 1125ndash1133 2005

[22] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[23] I Petras Fractional-Order Nonlinear Systems Springer BerlinGermany 2011

[24] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[25] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[26] G C Wu and D Baleanu ldquoDiscrete fractional logistic map andits chaosrdquo Nonlinear Dynamics 2013

[27] C P Li W H Deng and D Xu ldquoChaos synchronization of theChua system with a fractional orderrdquo Physica A vol 360 no 2pp 171ndash185 2006

[28] C Li and J Yan ldquoThe synchronization of three fractionaldifferential systemsrdquo Chaos Solitons amp Fractals vol 32 no 2pp 751ndash757 2007

[29] Z M Odibat N Corson M A Aziz-Alaoui and C BertelleldquoSynchronization of chaotic fractional-order systems via linearcontrolrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 1 pp 81ndash97 2010

[30] S Kuntanapreeda ldquoRobust synchronization of fractional-orderunified chaotic systems via linear controlrdquo Computers amp Math-ematics with Applications vol 63 no 1 pp 183ndash190 2012

[31] G Peng ldquoSynchronization of fractional order chaotic systemsrdquoPhysics Letters A vol 363 no 5-6 pp 426ndash432 2007

[32] L-G Yuan and Q-G Yang ldquoParameter identification andsynchronization of fractional-order chaotic systemsrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 305ndash316 2012

[33] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[34] H Zhu S Zhou and Z He ldquoChaos synchronization of thefractional-order Chenrsquos systemrdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2733ndash2740 2009

[35] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[36] J Yan and C Li ldquoOn chaos synchronization of fractionaldifferential equationsrdquo Chaos Solitons amp Fractals vol 32 no2 pp 725ndash735 2007

[37] S Bhalekar and V Daftardar-Gejji ldquoSynchronization of differ-ent fractional order chaotic systems using active controlrdquo Com-munications inNonlinear Science andNumerical Simulation vol15 no 11 pp 3536ndash3546 2010

[38] M M Asheghan M T H Beheshti and M S TavazoeildquoSynchronization and anti-synchronization of new uncertainfractional-order modified unified chaotic systems via novelactive pinning controlrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 2 pp 1044ndash1051 2011

[39] L Pan W Zhou L Zhou and K Sun ldquoChaos synchronizationbetween two different fractional-order hyperchaotic systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 6 pp 2628ndash2640 2011

[40] Z Odibat ldquoA note on phase synchronization in coupled chaoticfractional order systemsrdquoNonlinear Analysis Real World Appli-cations vol 13 no 2 pp 779ndash789 2012

[41] M Mahmoudian R Ghaderi A Ranjbar J Sadati S HHosseinnia and S Momani ldquoSynchronization of fractional-order chaotic system via adaptive PID controllerrdquo inNewTrendsin Nanotechnology and Fractional Calculus Applications pp445ndash452 Springer 2010

[42] L Jie L Xinjie and Z Junchan ldquoPrediction-control basedfeedback control of a fractional order unified chaotic systemrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 2093ndash2097 May 2011

[43] Y B Zhang W G Sun J W Wang and K Yang ldquoMixed syn-chronization for a class of fractionalorder chaotic systems viaopen-plus-closed-loop controlrdquo Communications in FractionalCalculus vol 3 pp 62ndash72 2012

[44] C Yin S-m Zhong and W-f Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[45] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[46] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[47] J Yuan S Bao and JWenqiang ldquoAdaptive slidingmode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[48] M R Faieghi H Delavari and D Baleanu ldquoControl of anuncertain fractional-order Liu system via fuzzy fractional-ordersliding mode controlrdquo Journal of Vibration and Control vol 18no 9 pp 1366ndash1374 2012

[49] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[50] J Yuan S Bao and D Chao ldquoPseudo-state sliding modecontrol of fractional SISO nonlinear systemsrdquo Advances inMathematical Physics In press

[51] C Yin S Dadras S-M Zhong and Y Chen ldquoControl of a novelclass of fractional-order chaotic systems via adaptive slidingmode control approachrdquo Applied Mathematical Modelling vol37 no 4 pp 2469ndash2483 2013

[52] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing Company2010

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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

Abstract and Applied Analysis 13

[53] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer London UK 2010

[54] J Lu G Chen D Cheng and S Celikovsky ldquoBridge the gapbetween the Lorenz system and the Chen systemrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 12 pp 2917ndash2926 2002

[55] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculusModels andNumericalMethods Series onComplexityNonlinearity and Chaos World Scientific Singapore 2012

[56] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

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

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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

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Algebra

Discrete Dynamics in Nature and Society

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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