synchronization of fractional order chaotic systems …...synchronization of fractional order...

$: Synchronization of fractional order chaotic systems …...Synchronization of fractional order chaotic systems using active control method S.K. Agrawal, M. Srivastava, S. Das Department$
Chaos, Solitons & Fractals 45 (2012) 737–752

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Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier .com/locate /chaos

Synchronization of fractional order chaotic systems using activecontrol method

S.K. Agrawal, M. Srivastava, S. Das ⇑Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi 221 005, India

a r t i c l e i n f o

Article history:Received 9 August 2011Accepted 4 February 2012Available online 14 March 2012

0960-0779/$ - see front matter � 2012 Elsevier Ltddoi:10.1016/j.chaos.2012.02.004

⇑ Corresponding author.E-mail address: [email protected] (S. Da

a b s t r a c t

In this article, the active control method is used for synchronization of two different pairsof fractional order systems with Lotka–Volterra chaotic system as the master system andthe other two fractional order chaotic systems, viz., Newton–Leipnik and Lorenz systemsas slave systems separately. The fractional derivative is described in Caputo sense. Numer-ical simulation results which are carried out using Adams–Bashforth–Moulton methodshow that the method is easy to implement and reliable for synchronizing the two nonlin-ear fractional order chaotic systems while it also allows both the systems to remain in cha-otic states. A salient feature of this analysis is the revelation that the time forsynchronization increases when the system-pair approaches the integer order from frac-tional order for Lotka–Volterra and Newton–Leipnik systems while it reduces for the otherconcerned pair.

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1. Introduction

The applications of dynamical systems include variousdisciplines of science and engineering. The concept of adynamical system has its origin in Newtonian mechanics.Mathematical model of a dynamical system is expressedeither in the form of a differential equation for a continu-ous time system or a difference equation, where time isdiscrete. The governing differential equation or the differ-ence equation mathematically expresses the present statesof the system in terms of the past ones for a constant set ofinputs and fixed parameters of the system. For a given setof initial conditions, it is possible to acquire knowledgeabout all the future states through solution of the govern-ing equation and determine the respective trajectory or or-bit, i.e., the graphical description of the states with respectto time.

Previously, modeling was primarily restricted to linearsystems for which analytical treatment is tractable. Re-cently due to the advent of powerful computers and withimproved computational techniques, it is possible to

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s).

tackle, to some extent, nonlinear systems. After all, nonlin-earity is an important feature of any real-time dynamicsystem and the study and analysis of non-linear dynamicshave gained immense popularity during the last fewdecades.

Chaotic systems are deterministic, nonlinear dynamicalsystems which are exponentially sensitive to initial condi-tions. This type of sensitivity is popularly known as thebutterfly effect [1]. The chaotic dynamics of fractional or-der systems is an important topic of study in nonlineardynamics. In the last few years this area of research hasbeen growing rapidly [2,3].

An example worth considering is the dynamical rela-tionship between predator and prey which is one of thedominant themes in ecology. It was observed from thepopulation data that interaction between a pair of preda-tor–prey influences the population growth of both the spe-cies. The first theoretical treatment of population dynamicswas presented by Malthus [4], viz., ‘‘Essay on the principleof population’’. Verhulst [5] formed a mathematical modelbased on ‘‘principle of population’’, viz., ‘‘The logistic equa-tion’’. The next major advancement in population dynam-ics was presented by Lotka [6] and Volterra [7]. Theypresented for the first time the differential equations of

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738 S.K. Agrawal et al. / Chaos, Solitons & Fractals 45 (2012) 737–752

predator–prey type (tropic interaction model). Since thenmore complicated but realistic predator–prey systemshave been used by ecologists and mathematicians. In1988, Samardzija and Greller [8] proposed a two-predator,one prey generalization of the Lotka–Volterra equations.The proposed three dimensional system exhibits chaoticbehavior, takes an explosive route to chaos and in variousregions of parameter space evolves on a fractal torus.

In 1981, Newton and Leipnik [9] obtained the set of dif-ferential equations from Euler rigid body equations whichwere modified with the addition of a linear feedback. Thisis useful to model dynamics conflict with mediation, e.g.,conflict between townspeople, university students and po-lice. For a certain set of parameters the system exhibitschaotic behavior and displays the existence of two strangeattractors. An attractor is a state point toward which thetrajectories of a dynamic system would normally tend tomove.

The Lorenz attractor is an example of a non linear dy-namic system corresponding to the long term behavior ofthe Lorenz oscillation. The Lorenz oscillator is a threedimension dynamical system that exhibits lemniscatestype shaped chaotic flow which shows how the state ofdynamical system evolves over time in a complex, non-repeating pattern. The Lorenz equations [10] deal withthe stability of fluid flows in the atmosphere. In additionto its interest in the field of non linear mathematics, theLorenz model has important implications for climate andweather prediction. The case is also applicable for simpli-fied models for lasers [10] and dynamos [11].

The concept of fractional derivative is used increasinglyto model the behavior of real systems in various fields ofscience and engineering including Fluid Mechanics[12,13], Viscoelasticity [14], Material Science [15], Bioengi-neering [16], Medicine [17], Biological Tissues [18], CardiacTissue [19], etc. The attribute of fractional order systemsfor which they have gained popularity in the investigationof dynamical systems is that they allow a greater degree offlexibility in the model. An integer order differential oper-ator is a local operator. Whereas the fractional order differ-ential operator is non local in the sense that it takes intoaccount the fact that the future state not only dependsupon the present state but also upon all of the history ofits previous states. For this realistic property, the usageof fractional order systems is becoming popular. It is tobe noted that the present states of any real-life dynamicsystem are dependent upon the history of its past states.The memory element is very much relevant.

The fractional operators can deal with systems withlong-range time correlations, which are related to the slowdecay of the inverse power-law kernel in the fractionaloperators. The evolution equations are obtained by replac-ing the first order derivative by a fractional orderqð0 < q 6 1). Gorenflo and Mainardi [20] in their articlehave shown how such evolution equations can be obtainedfrom a more general master equations which govern socalled continuous time random walk (CTRW) by a properlyscaled passage to the limit of compressed waiting timesand jump width. CTRW is a combination of random walkon the axis of physical time and a random walk in space.Recent survey of a most efficient model, viz., anomalous

diffusion equation which is based on the concept of CTRWexhibits random waiting times between two successivejumps. Fractional time or space diffusion equations are rel-evant in this case for the limiting dynamics of CTRW.Gorenflo et al. [21] demonstrated that for obtaining thespace–time fractional diffusion equation describing diffu-sion process, the essential assumption is that the probabil-ities for waiting times and jump widths behaveasymptotically like powers with negative exponents re-lated to the orders of the fractional derivatives. In 2001,Paradisi et al. [22] have proposed a general Fick’s law usingfractional derivative operator which accounts for non-localphenomena by virtue of its integral nature. In their otherarticle [23], they have presented a relationship betweenflux and concentration gradient during their study of ageneralized non-local Fick’s law derived from the frac-tional diffusion equation generating the Levy–Feller statis-tics. Recent study on fractional derivative reveals that itcan behave as a dissipative term. Fa [24] has made thisobservation through the analysis of simple fractionaloscillator.

In the last decade, fractional order modeling has beenan active field of research both from the theoretical andthe applied perspectives. A wide range of problems in dif-ferent branches of engineering and biology have alreadybeen studied by a number of researchers of different partsof the world to explore the potential of the fractional deriv-ative. The usage of first order time derivative with a frac-tional order time derivative is not only applicable fornon-Gaussian but also for non-Markovian systems. Re-cently Hanert et al. [25] have considered the propagationof epidemic fronts during their study of fractional orderepidemic model and have shown that the use of fractionalorder diffusion term can estimate the acceleration of frontthat causes a rapid spread of epidemic much faster thanpredicted by classical Gaussian model. Using stability anal-ysis on a fractional order model of HIV infection of CD4+ T-cells, Ding and Ye [26] obtained a sufficient condition onthe parameters for the stability of the infected steady state.The area of fractional order dynamical systems viz., Lotka–Volterra, Newton–Leipnik, Lorenz models have alreadybeen explored to a certain extent. In 2007, Ahmed et al.[27] considered the fractional order predator-prey modeland the fractional order rabies model. In 2009, Das et al.[28] have solved a fractional order Lotka–Volterra modelwith one prey and one predator. In 2010, Das and Gupta[29] solved the same type of more general model consider-ing the growth rate of the prey, the efficiency of predator,the growth rate of predator and the death rate of predatoras functions of time. A fractional model of two preys andone predator can be found in the book of Petras [30]. Thedynamics of the fractional order Lorenz model was alsostudied by Grigorenko and Grigorenko [31]. Based on thequalitative theory, the existence and uniqueness of solu-tions for a class of fractional-order Lorenz chaotic systemshave been investigated theoretically by Yu et al. [32]. Frac-tional order diffusion less Lorenz system has been investi-gated numerically by Sun and Sprott [33] and results haveshown that the system has complex dynamics with inter-esting characteristics. In 2008, Kang et al. [34] studiedthe influence of parameters on the dynamics of a fractional

S.K. Agrawal et al. / Chaos, Solitons & Fractals 45 (2012) 737–752 739

order Newton–Leipnik system and showed that the systemdisplays comprehensive dynamical behaviors like fixedpoints, periodic motion, chaotic motion, etc. Diethelmet al. [35] used predictor-corrector scheme for solving frac-tional order Newton–Leipnik system. Recently, Zhang et al.[36] have studied the stability analysis of the fractional or-der Newton–Leipnik system using fractional Routh–Hurwitz criteria.

Synchronization of chaos is a phenomenon that may oc-cur when two or more chaotic systems are coupled or onechaotic system drives the other. The pioneering work ofPecora and Carroll [37] introduced a method about syn-chronization between the drive (master) and response(slave) systems of two identical or non identical systemswith different initial conditions, which has attracted agreat deal of interest in various fields due to its importantapplications in ecological systems [38], physical systems[39], chemical systems [40], modeling brain activity, sys-tem identification, pattern recognition phenomena and se-cure communications [41,42], etc. In recent years varioussynchronization schemes, such as linear and nonlinearfeedback synchronization [43,44], time delay feedback ap-proach [45], adaptive control [46], active control, [47,48],etc. have been successfully applied to chaos synchroniza-tion. However, most of the methods mentioned above havesynchronized two identical chaotic systems. We, theauthors, have made an endeavor to study and analyze thesynchronization two different types or different pairs offractional order systems. We started off considering thetypical prey–predator equation in ecological modeling fol-lowed by a thorough investigation of work done byresearchers in the area during the last decade. In 2002,Ho and Hung [47] successfully applied the active controlmethod for synchronization of two different chaotic sys-tems, viz., easy periodic and Rossler systems. In 2006, Park[49] investigated chaos synchronization between two dif-ferent chaotic systems, viz., Genesio system and Rosslersystem using nonlinear control laws, taking first systemas a master and second one as slave. In 2007, Yan and Li[50] presented chaos synchronization of fractional orderLorenz, Rossler and Chen systems taking one as masterand second one as slave. In 2008, Vincent [51] made anexcellent effort for examining chaos synchronization be-tween two nonlinear systems using two different tech-niques viz., active control and back stepping control interms of transient analysis. In 2008, Zhou and Cheng [52]synchronized between different fractional order chaoticsystems viz., Rossler and Chen systems and Chua and Chensystems. Synchronization between chaotic systems is alsobeing widely investigated by Refs. [50,53,54]. Recently,Faieghi and Delavari [55] and Sundarapandian [56] havesuccessfully applied the active control method for synchro-nization of identical and different chaotic systems, respec-tively. But to the best of authors’ knowledge thesynchronization of fractional order Lotka–Volterra systemwith fractional order Newton–Leipnik and Lorenz systemshave not yet been studied by any researcher. The theme ofthe study is to investigate the time required for synchroni-zation between two different pair of chaotic systems whenthe time derivative changes to fractional order from thestandard one.

In this article the authors have done the synchroniza-tion between two different pairs of fractional order chaoticsystems namely the Lotka–Volterra and Newton–Leipniksystems and the Lotka–Volterra and Lorenz systems usingactive control method. Using Adams–Bashforth–Moultonmethod (see Appendix A), numerical simulations are car-ried out for different order fractional derivatives and forstandard one which are depicted through graphs for differ-ent particular cases.

2. Synchronization between different fractional orderchaotic systems

Fractional calculus is a generalization of integration anddifferentiation to a non integer order integro-differentialoperator aDq

t is defined by

aDqt ¼

dq

dtq ; RðqÞ > 0;1; RðqÞ ¼ 0;R t

aðdsÞ�q RðqÞ < 0;

8><>: ð1Þ

where q is the fractional order which can be a complexnumber, R(q) denotes the real part of q and a < t, a is thefixed lower terminal and t is the moving upper terminal.

There are some definitions for fractional derivative. Thecommonly used definition is Riemann–Liouville definition,defined by

aDqt xðtÞ ¼ dn

dtn jn�qt xðtÞ; q > 0; ð2Þ

n ¼ dqe i.e., n is the first integer which is not less than q; jatis the a-order Riemann–Liouville integral operator which isdescribed as follows:

jat uðtÞ ¼1

CðaÞ

Z t

0

uðsÞðt � sÞ1�a ds; ð3Þ

where 0 < a 6 1 and C(�) is gamma function.In our work the following definition is used:

Dqt xðtÞ ¼ jn�q

t xðnÞðtÞ; q > 0; ð4Þ

where n ¼ dqe and the operator Dqt is the Caputo differ-

ential operator of order q. For an initial problem ofRiemann–Liouville type, one would have to specify thevalues of certain fractional derivatives (and integrals) ofunknown solution at the initial point t = 0. However, it isnot clear what the physical meanings of fractional deriva-tives of x are when we deal with a concrete physical appli-cation, and hence it is also not clear how such quantitiescan be measured. When we deal with an initial problemof Caputo type, the situation is different. We may specifythe initial values. This clearly helps to understand thephysical meaning.

Again Riemann–Liouville initial value problems requirehomogeneous initial conditions though for Caputo initialvalue problem holds for both homogeneous and non-homogeneous conditions. For this reason choice derivativeis better than the Riemann–Liouville derivative.

The important reason of choosing Caputo derivatives forsolving initial value fractional order differential equationsis that it does not introduce any problems when trying to


model real world phenomena with fractional time andapace derivatives which are noticed for Riemann–Liouvillederivative.

Another important difference between the Riemann–Liouville definition and Caputo definition is that the Cap-uto derivative of a constant is zero, whereas in the caseof the Riemann–Liouville fractional derivative of the con-stant c is not equal to zero but 0Dq

t c ¼ c t�q

Cð1þqÞ :

Let us consider the two different fractional order cha-otic systems

dqxdtq ¼ Axþ FðxÞ; ð5Þ

dqydtq ¼ Byþ GðyÞ; ð6Þ

where 0 < q 6 1 is the order of the fractional time deriva-tive, x, y 2 Rn are the state vectors, FðxÞ;GðyÞ 2 Rn are thenonlinear terms of the system (5) and system (6), and aresmooth functions. A, B are constant matrices. For synchro-nization of the above two different systems, system (5)represents the master (drive) system and system (6) repre-sents the slave (response) system.

Now introducing the active control parameter u 2 Rn inthe system (6), we get.

dqydtq ¼ Byþ GðyÞ þ u: ð7Þ

The purpose of chaos synchronization is how to design theactive controller u, which is able to synchronize the statesof both the master and the slave systems. If we define theerror vector as e = y � x, the error dynamical systembecomes

dqedtq ¼ Beþ ðB� AÞxþ GðyÞ � FðxÞ þ u: ð8Þ

Theorem. Suppose

u ¼ ðA� BÞxþ FðxÞ � GðxÞ þ DGðxÞeþ Ce; ð9Þ

where DG(x) is the Jacobi matrix of G(x). C is a controller gainmatrix to be designed later.

If j argðkiðBþ CÞÞj > 0:5pq; i ¼ 1;2; . . . ;n by choosingsuit controller gain matrix control C, then the fractional-or-der chaotic system (5) and the fractional-order chaotic sys-tem (7) can arrive to the asymptotic synchronization. HereargðkiðBþ CÞÞ denotes the argument of the eigenvalues ki of(B + C).

Proof. The proof of above theorem can be found inAppendix B with the help of [52]. h

2.1. The fractional-order Lotka–Volterra system

The fractional-order Lotka–Volterra system (in the arti-cle 5.13 from the book by Petras [30]) is given by

dq1 xdtq1

¼ ax� bxyþ ex2 � szx2;

dq2 ydtq2

¼ �cyþ dxy;

dq3 zdtq3

¼ �pzþ szx2;

ð10Þ

where parameters a, b, c, d > 0, ‘a’ represents the naturalgrowth rate of the prey in the absence of predators, ‘b’ rep-resents the effect of predation on the prey, ‘c’ representsthe natural death rate of the predator in the absence ofprey, ‘d’ represents the efficiency and propagation rate ofthe predator in the presence of prey, and e, p, s are positiveconstants.

During synchronization the parameters are taken asa = 1, b = 1, c = 1, d = 1, e = 2, s = 2.7, p = 3 and the initialcondition is [1,1.4,1], 0 < qi 6 1 is the order of the deriva-tive. At qi = 0.95 (i = 1, 2, 3), Eq. (10) represents the frac-tional order Lotka–Volterra chaotic system and thechaotic attractors of the system (10) are described throughFig. 1.The phase portraits in x–y–z space and x–y, y–z, z–xplanes are shown through Fig. 1(a)–(d) respectively.

2.2. The fractional-order Newton–Leipnik system

The fractional-order Newton–Leipnik system [57] is gi-ven by

dq1 xdtq1

¼ �axþ yþ 10yz;

dq2 ydtq2

¼ �x� 0:4yþ 5xz;

dq3 zdtq3

¼ bz� 5xy;

ð11Þ

where ‘a’ and ‘b’ are variable parameters. Usually theparameter ‘b’ is taken in the interval (0,8.0). The systemis ill-behaved outside this interval. As b ? 0, the systemrelatively shows uninteresting dynamics and for b P 0:8,the given system becomes explosive i.e., the solution di-verge to infinity for any initial condition other than thecritical points.

During synchronization the parameters are taken asa = 0.4, b = 0.175, initial condition = [0.19 0 �0.18] and0 < qi 6 1 is order of derivative. At qi = 0.95 (i = 1, 2, 3),Eq. (11) becomes the fractional order Newton–Leipnik cha-otic equation, and the chaotic attractors of fractional ordersystem (11) are described in Fig. 2. The phase portraits inx–y–z space and x–y, y–z, z–x planes are shown throughFig. 2(a)–(d) respectively.

2.3. The fractional-order Lorenz system

The fractional-order Lorenz system [31,58] is given by

dq1 xdtq1

¼ rðy� xÞ;

dq2 ydtq2

¼ xðq� zÞ � y;

dq3 zdtq3

¼ xy� bz;

ð12Þ

00.5

11.5

22.5

0.51

1.52

2.50

0.5

1

1.5

2

2.5

x(t)y(t)

z(t)

0 0.5 1 1.5 2 2.50.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

x(t)

y(t)

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60

0.5

1

1.5

2

2.5

y(t)

z(t)

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

x(t)

z(t)

1(a) 1(b)

1(c) 1(d)

Fig. 1. Phase portraits of Lotka–Volterra system in x–y–z space and x–y, y–z, x–z planes.


where r is the Prandtl number, q is the Rayleigh numberand l is the size of the region approximated by the system.

During synchronization the parameters are taken asr = 10, q = 28, l = 8/3, the initial conditions = [0.1, 0.1,0.1], and 0 < qi 6 1 is order of derivative. When qi = 0.993(i = 1,2,3), Eq. (12) represents the fractional order Lorenzchaotic equation and the chaotic attractors of fractional or-der system (12) are described in Fig. 3. The phase portraitsin x–y–z space and x–y, y–z, z–x planes are shown throughFig. 3(a)–(d) respectively.

2.4. Synchronization of fractional order Lotka–Volterra andNewton–Leipnik systems via active control method

In this section the synchronization behavior betweentwo different fractional orders Lotka–Volterra andNewton–Leipnik is made. We assume that Lotka–Volterrasystem drives the Newton–Leipnik system. Therefore, we

define the Lotka–Volterra as a master system and New-ton–Leipnik as a slave system as follows:

The master system is described by (10) as

dq1 x1dtq1 ¼ a1x1 � b1x1y1 þ Ex2

1 � sz1x21;

dq2 y1dtq2 ¼ �c1y1 þ d1x1y1;

dq3 z1dtq3 ¼ �pz1 þ sz1x2

1:

8>><>>:

ð13Þ

The slave system is described by (11) as

dq1 x2dtq1 ¼ �a2x2 þ y2 þ 10y2z2 þ u1ðtÞ;

dq2 y2dtq2 ¼ �x2 � 0:4y2 þ 5x2z2 þ u2ðtÞ;

dq3 z2dtq3 ¼ b2z2 � 5x2y2 þ u3ðtÞ;

8>><>>:

ð14Þ

where three active control functions u1(t), u2(t) and u3(t)are introduced in Eq. (14). Our goal is to investigate thesynchronization of system (13) and (14). We define the er-

-0.3-0.2

-0.10

0.10.2

-0.4-0.2

00.2

0.4-0.3

-0.2

-0.1

0

0.1

x(t)y(t)

z(t)

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

x(t)

y(t)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

y(t)

z(t)

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

x(t)

z(t)

2(b)2(a)

2(c) 2(d)

Fig. 2. Phase portraits of Newton–Leipnik system in x–y–z space and x–y, y–z, x–z planes,


ror states e1 = x2 � x1, e2 = y2 � y1, e3 = z2 � z1. The corre-sponding error dynamics can be obtained by subtractionof Eq. (13) from Eq. (14)

dq1 e1dtq1 ¼�a2e1�ða1þa2Þx1þe2þy1�Ex2

1þb1x1y1þsz1x21þ10y2z2þu1ðtÞ;

dq2 e2dtq2 ¼�e1�0:4e2�x1þðc1�0:4Þy1�d1x1y1þ5x2z2þu2ðtÞ;

dq3 e3dtq3 ¼b2e3þðb2þpÞz1�sz1x2

1�5x2y2þu3ðtÞ:

8>><>>:

ð15Þ

Then we define the active control inputs u1(t), u2(t) andu3(t) as

u1ðtÞ¼ v1ðtÞþða1þa2Þx1�y1þEx21�b1x1y1� sz1x2

1�10y2z2;

u2ðtÞ¼ v2ðtÞþx1�ðc1�0:4Þy1þd1x1y1�5x2z2;

u3ðtÞ¼ v3ðtÞ�ðb2þpÞz1þ sz1x21þ5x2y2;

8><>:

ð16Þ

which leads to

dq1 e1dtq1 ¼ �a2e1 þ e2 þ v1ðtÞ;

dq2 e2dtq2 ¼ �e1 � 0:4e2 þ v2ðtÞ;

dq3 e3dtq3 ¼ b2e3 þ v3ðtÞ:

8>><>>:

ð17Þ

The synchronization error system (17) is a linear systemwith active control inputs, v1(t), v2(t) and v3(t). Next designan appropriate feedback control which stabilizes the sys-tem so that e1, e2, e3 converge to zero as time t tends toinfinity, which implies that Lotka–Volterra and Newton–Leipnik system are synchronized with feedback control.There are many possible choices for the control inputsv1(t), v2(t) and v3(t). We choose

-20-10

010

20

-40-20

020

400

10

20

30

40

50

x(t)y(t)

z(t)

-20 -15 -10 -5 0 5 10 15 20-30

-20

-10

0

10

20

30

x(t)

y(t)

-30 -20 -10 0 10 20 300

5

10

15

20

25

30

35

40

45

50

y(t)

z(t)

-20 -15 -10 -5 0 5 10 15 200

5

10

15

20

25

30

35

40

45

50

x(t)

z(t)

3(c) 3(d)

3(a) 3(b)

Fig. 3. Phase portraits of Lorenz system in x–y–z space and x–y, y–z, x–z planes.


v1ðtÞ

v2ðtÞ

v3ðtÞ

2664

3775 ¼ C

e1

e2

e3

2664

3775; ð18Þ

where C is a 3 � 3 constant matrix. In order to make theclosed loop system stable, the matrix C should be selectedin such a way that the feedback system has eigenvalues ki

of C satisfies the control j argðkiÞj > 0:5pq; i ¼ 1;2;3:Thereis not a unique choice for such matrix C, a good choice canbe as follows:

C ¼

a2 � 1 �1 0

0 �0:6 0

0 0 �b2 � 1

0BB@

1CCA: ð19Þ

Then the error system is changed to

dq1 e1dtq1 ¼ �e1;

dq2 e2dtq2 ¼ �e1 � e2;

dq3 e3dtq3 ¼ �e3:

8>>><>>>:

ð20Þ

All the three eigenvalues of the system are �1, hence theconditions that all qi 6 1 are satisfied and we get the re-quired synchronization.

2.5. Simulation and results

In numerical simulations, the parameters of the New-ton–Leipnik system are taken as a2 = 0.4 and b2 = 0.175.Parameters of the Lotka–Volterra system are taken asa1 = 1, b1 = 1, c1 = 1, d1 = 1, E = 2, p = 3, s = 2.9851. Time stepsize was taken as 0.005. The initial values of the mastersystem (Lotka–Volterra system) and the slave system(Newton–Leipnik system) are taken as [1,1.4,1] and[0.349,0,-0.160] respectively. Thus, the initial errors are[�0.651,�1.4,�1.160].

Figs. 4–6 demonstrate that systems are synchronizedafter small duration of time for the considered fractionalorder time derivatives qi = 0.80, qi = 0.95 and also for stan-dard case qi = 1, i = 1, 2, 3. It is also seen from the figuresthat the time taken for synchronization of two systems in-creases with the increase of fractional orders approachingtowards standard order system.

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

t

x 1(t), x

2(t)

x1 (t)

x2 (t)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

t

y 1(t), y

2(t)

y 1 (t)

y2 (t)

4(a)

4(b)

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

z 1(t), z

2(t)

z1 (t)

z2 (t)

0 1 2 3 4 5 6 7 8 9 10-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

e 1(t), e

2(t), e

3(t)

t

e1 (t)

e2 (t)

e3 (t)

4(c)

4(d)

Fig. 4. State trajectories of drive system (13) and response system (14) for the fractional order qi = 0.80 (i = 1,2,3): (a) between x1 and x2, (b) between y1 andy2, (c) between z1 and z2, (d) The evolution of the error functions e1(t), e2 (t) and e3(t).


2.6. Synchronization of fractional order Lotka–Volterra andLorenz systems via active control method

In this section the synchronization behavior in twodifferent fractional order Lotka–Volterra and Lorenz sys-

tems is made. We assume that Lotka–Volterra systemdrives the Lorenz system. Therefore, we define theLotka–Volterra as a master system and Lorenz as a slavesystem.

The master system is given by (10)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

t

y 1(t), y

2(t)

y1 (t)y2 (t)

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

z 1(t), z

2(t)

z 1(t)z2(t)

0 1 2 3 4 5 6 7 8 9 10-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

e 1(t), e

2(t), e

3(t)

t

e 1 (t)e2 (t)e3 (t)

0 1 2 3 4 5 6 7 8 9 100.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

x 1(t), x

2(t)

x 1(t)x2(t)

5(a)

5(b)

5(c)

5(d)

Fig. 5. State trajectories of drive system (13) and response system (14) for the fractional order qi = 0.95 (i = 1,2,3): (a) between x1 and x2, (b) between y1 andy2, (c) between z1 and z2, (d) the evolution of the error functions e1(t), e2(t) and e3(t).


dq1 x1dtq1 ¼ ax1 � bx1y1 þ Ex2

1 � sz1x21;

dq2 y1dtq2 ¼ �cy1 þ dx1y1;

dq3 z1dtq3 ¼ �pz1 þ sz1x2

1:

8>><>>:

ð21Þ

The slave system is described by (12)

dq1 x2dtq1 ¼ rðy2 � x2Þ þ u1ðtÞ;

dq2 y2dtq2 ¼ x2ðq� z2Þ � y2 þ u2ðtÞ;

dq3 z2dtq3 ¼ x2y2 � bz2 þ u3ðtÞ:

8>>>>>><>>>>>>:

ð22Þ

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

t

x 1(t),x

2(t)x 1 (t)x2(t)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

y 1(t),y

2(t)

y1 (t)y2(t)

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

2

2.5

t

z 1(t),z

2(t)

z 1(t)z2(t)

0 1 2 3 4 5 6 7 8 9 10-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

t

e 1(t),e

2(t),e

3(t)

e 1 (t)e 2 (t)e 3 (t)

6(d)

6(c)

6(d)

6(a)

Fig. 6. State trajectories of drive system (13) and response system (14) for the standard order qi = 1(i = 1,2,3): (a) between x1 and x2, (b) between y1 and y2,(c) between z1 and z2, (d) the evolution of the error functions e1(t), e2(t) and e3(t).


We have introduced three active control functions u1(t),u2(t) and u3(t) in (22). Our goal is to investigate the syn-chronization of the systems (21) and (22). We define theerror states e1 = x2 � x1, e2 = y2 � y1, e3 = z2 � z1. The corre-sponding error dynamics can be obtained by subtractionEq. (21) from Eq. (22) as

dq1 e1dtq1 ¼re2þry1þae1�ðaþrÞx2�Ex2

1þbx1y1þsz1x21þu1ðtÞ;

dq2 e2dtq2 ¼�ce2þðc�1Þy2�dx1y1þqx2�x2z2þu2ðtÞ;

dq3 e3dtq3 ¼�be3þðp�bÞz1�sz1x2

1þx2y2þu3ðtÞ:

8>>>><>>>>:

ð23Þ


Then we define the active control inputs u1(t), u2(t), andu3(t) as

0 2 4 6 8 100.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

y 1(t), y

2(t)

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

t

z 1(t), z

2(t)

0 2 4 6 8 100

2

4

6

8

10

12

t

e 1(t),e

2(t),e

3(t)

7(

7(

7(

0 2 4 6 8 10

2

4

6

8

10

12

t

x 1(t), x

2(t)

7(

Fig. 7. State trajectories of drive system (21) and response system (22) for the fray2, (c) between z1 and z2, (d) the evolution of the error functions e1(t), e2(t) and

u1ðtÞ ¼ v1ðtÞ þ ðaþ rÞx2 � ry1 þ Ex21 � bx1y1 � sz1x2

1;

u2ðtÞ ¼ v2ðtÞ � qx2 � ðc � 1Þy2 þ dx1y1 þ x2z2;

u3ðtÞ ¼ v3ðtÞ � ðp� bÞz1 þ sz1x21 � x2y2:

8><>:

ð24Þ

12 14 16 18 20

y1 (t)

y2 (t)

12 14 16 18 20

z1(t)

z2 (t)

12 14 16 18 20

e 1 (t)

e 2 (t)

e 3 (t)

b)

c)

d)

0 12 14 16 18 20

x 1 (t)

x 2 (t)

a)

ctional order qi = 0.95 (i = 1,2,3): (a) between x1 and x2, (b) between y1 ande3(t).


which leads to

dq1 e1dtq1 ¼ e1 þ re2 þ v1ðtÞ;

dq2 e2dtq2 ¼ �ce2 þ v2ðtÞ;

dq3 e3dtq3 ¼ �be3 þ v3ðtÞ:

8>><>>:

ð25Þ

0 2 4 6 8 10

5

10

15

t

x 1(t),

x 2(t)

0 2 4 6 8 10.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

y 1(t),

y 2(t)

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

10

t

z 1(t),

z 2(t)

0 2 4 6 8 10

2

4

6

8

10

12

14

t

e 1(t),

e 2(t),

e 3(t)

8(

8(

8(

8(

Fig. 8. State trajectories of drive system (21) and response system (22) for the fand y2, (c) between z1 and z2, (d) the evolution of the error functions e1(t), e2(t)

The synchronization error system (25) is a linear systemwith active control inputs, v1(t), v2(t) and v3(t). Next to de-sign an appropriate feedback control stabilizes the systemso that e1, e2, e3 converge to zero as time t tends to infinity,which implies that Lotka–Volterra and Lorenz system are

0 12 14 16 18 20

x1

(t)

x2

(t)

0 12 14 16 18 20

y1

(t)

y2

(t)

0 12 14 16 18 20

z1

(t)

z2

(t)

0 12 14 16 18 20

e1

(t)

e2

(t)

e3

(t)

a)

b)

c)

d)

ractional order qi = 0.993 (i = 1,2,3): (a) between x1 and x2, (b) between y1

and e3 (t).


synchronized with feedback control. As previous one, wechoose

v1ðtÞv2ðtÞv3ðtÞ

264

375 ¼ C

e1

e2

e3

264

375; ð26Þ

0 2 4 6 8 100

5

10

15

t

x 1(t),x

2(t)

0 2 4 6 8 100.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

y 1(t),y

2(t)

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

t

z 1(t),z

2(t)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

t

e 1(t),e

2(t),e

3(t)

9(a

9(b

9(c

9(

Fig. 9. State trajectories of drive system (21) and response system (22) for the sta(c) between z1 and z2, (d) the evolution of the error functions e1(t), e2(t) and e3(

where C is a 3 � 3 constant matrix. In order to make theclosed loop system stable, the matrix C should be selectedin such a way that the feedback system has eigenvalues ki

of C satisfies the control j argðkiÞj > 0:5pq; i ¼ 1;2;3. Let uschoose the matrix C as

12 14 16 18 20

x1(t)

x2 (t)

12 14 16 18 20

y1(t)

y2 (t)

12 14 16 18 20

z1(t)

z2(t)

12 14 16 18 20

e1 (t)

e2 (t)

e3 (t)

)

)

)

d)

ndard order qi = 1 (i = 1,2,3): (a) between x1 and x2, (b) between y1 and y2,t).


C ¼�2 0 00 c � 1 00 0 b� 1

0B@

1CA; ð27Þ

then the error system is changed as

dq1 e1dtq1 ¼ �2e1 þ re2;

dq2 e2dtq2 ¼ �e2;

dq3 e3dtq3 ¼ �e3:

8>><>>:

ð28Þ

Here also all the three eigenvalues of the system are �1,hence the condition that all qi 6 1 is satisfied and hencewe get the required synchronization.

2.7. Simulation and results

In numerical simulations, the parameters of the Lorenzsystem are taken as r = 10, q = 28 and b = 8/3. Parametersof the Lotka–Volterra system are taken as a = 1, b = 1, c = 1,d = 1, E = 2, p = 3, s = 2.9851. Time step size is taken as0.005. The initial values of the master system (for Lotka–Volterra system) and the slave system (Lorenz system)are taken as [1,1.4,1] and [10 5 10] respectively. Thus,the initial errors are [9,3.6,9].

Fig. 7 demonstrates that it takes higher time for syn-chronization of two considered chaotic systems for thefractional order qi = 0.95 (i = 1,2,3) in comparison to thesynchronization for the fractional order qi = 0.993 whichis displayed through Fig. 8 and also for standard order dis-played through Fig. 9. This shows that for this pair of frac-tional order systems it takes lesser synchronization timewith the increase of fractional orders while approachingtowards standard order system.

3. Conclusion

There are three important goals that the authors haveachieved in the present article. First one is the study ofdynamical behavior of coupled chaotic systems of frac-tional order. Second one is employing the powerful activecontrol method which provides us a simple way to syn-chronize a pair of chaotic systems. Thirdly, the observationthat the synchronization time increases when the systempair approaches the standard order from fractional orderwith Lotka–Volterra as the master & Newton–Leipnik asthe slave while it reduces when Lotka–Volterra drives theLorenz system is a major outcome of the study. Numericalsimulations are used to verify the efficiency, effectivenessand validity of the proposed method. It is worth mention-ing that since synchronization of two fractional order cha-otic systems assumes considerable significance in thestudy of nonlinear dynamics, the outcome of this researchwork would be appreciated and could be utilized by thoseresearchers involved in the field of mathematical modelingof fractional order dynamical systems.

Acknowledgements

The authors of this article express their heartfelt thanksto the reviewers for their valuable suggestions for the

improvement of the article. The first author acknowledgesthe financial support from the UGC, New Delhi, India, underthe JRF scheme. The corresponding author Dr. S. Das ex-presses his heartfelt thanks to Prof. Ong S.H., Institute ofMathematical Science, University of Malaya, Malaysia forconsidering him as Academic Visitor to the institute.

Appendix A

Numerical methods used for solving ODEs have to bemodified for solving fractional differential equations (FDEs).We only derive the predictor–corrector scheme fordrive–response systems. This scheme is the generalizationof Adams–Bashforth–Moulton one [59,60]. We interpretthe approximate solution of nonlinear fractional-order dif-ferential equations using this algorithm in the followingway.

The considered differential equation

Dqt yðtÞ ¼ f ðt; yðtÞÞ; 0 6 t 6 T; ðA1Þ

with

yðkÞð0Þ ¼ yðkÞ0 ; k ¼ 0;1; . . . ; ½q�

is equivalent to the Volterra integral equation

yðtÞ ¼X½q��1

k¼0

yðkÞ0tk

k!þ 1

CðqÞ

Z t

0ðt � sÞq�1f ðs; yðsÞÞds: ðA2Þ

Set, h = T/N, tn = nh, n = 0,1, . . ., ,N 2 Z+. Then (A2) can be dis-credited as follows:

yhðtnþ1Þ ¼X½q��1

k¼0

yðkÞ0

tknþ1

k!þ hq

Cðqþ 2Þ f ðtnþ1; yphðtnþ1ÞÞ

þ hq

Cðqþ 2ÞXn

j¼0

aj;nþ1f ðth; yhðtjÞÞ; ðA3Þ

aj;nþ1¼

nqþ1�ðn�qÞðnþ1Þq if j¼0;

ðn� jþ2Þqþ1þðn� jÞqþ1�2ðn� jþ1Þqþ1 if 06 j6n;

1 if j¼nþ1;

8>>><>>>:

ðA4Þ

where predicted value yh(tn+1) is determined by

yphðtnþ1Þ ¼

X½q��1

k¼0

yðkÞ0

tknþ1

k!þ 1

CðqÞXn

j¼0

bj;nþ1f ðtj; yhðtjÞÞ; ðA5Þ

bj;nþ1 ¼hq

qððnþ 1� jÞq � ðn� jÞqÞ: ðA6Þ

The error estimate is

maxj¼0;1...N

jyðtjÞ�yhðtjÞj¼oðhpÞ in which p¼minð2;1þqÞ: ðA7Þ

Appendix B

From Eqs. (8) and (9), we have

dqedtq ¼ Beþ GðyÞ � GðxÞ � DGðxÞeþ Ce: ðB1Þ


On the other hand, using Taylor formula in the neighbor-hood UðxkekÞ we derive that

GðyÞ ¼ GðxÞ þ DGðxÞeþ OðkekÞ: ðB2Þ

So, combining (B1) and (B2), we obtain

dqedtq ¼ ðBþ CÞeþ OðkekÞ: ðB3Þ

Certainly, ei = yi � xi = 0, i = 1,2,3, . . . ,n is one fixed point oferror dynamical system (B3). The Jacobi matrix of (B3) atthis fixed point is (B + C). As shown by D. Matignon [61]for analyzing the stability criterion for the fractional differ-ential equation, since the argument of the eigenvalues ki ofmatrix (B + C) satisfy j argðkiðBþ CÞÞj > 0:5pq; thereforethis fixed point considered here is asymptotically stable.

Now, limt!1ei ¼ 0 ði ¼ 1;2; . . . ;nÞ, indicates that sys-tems (7) and (5) can be achieved to synchronization. Thusthe synchronization condition is that all the eigenvalues ki

of matrix (B + C) in (B3) satisfy j argðkiðBþ CÞÞj > 0:5pq ði ¼1;2;3Þ.

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http://dx.doi.org/10.1016/j.cnsns.2011.05.03

synchronization of fractional order chaotic systems …...synchronization of fractional order...

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