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International Journal of Differential Equations
Volume 2011 (2011), Article ID 250763, 13 pages
http://dx.doi.org/10.1155/2011/250763
Research Article

Antisynchronization of Nonidentical Fractional-Order Chaotic Systems Using Active Control

1Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur 416004, India
2Department of Mathematics, University of Pune, Ganeshkhind, Pune 411007, India

Received 7 May 2011; Accepted 16 July 2011

Academic Editor: Wen Chen

Copyright © 2011 Sachin Bhalekar and Varsha Daftardar-Gejji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Antisynchronization phenomena are studied in nonidentical fractional-order differential systems. The characteristic feature of antisynchronization is that the sum of relevant state-variables vanishes for sufficiently large value of time variable. Active control method is used first time in the literature to achieve antisynchronization between fractional-order Lorenz and Financial systems, Financial and Chen systems, and Lü and Financial systems. The stability analysis is carried out using classical results. We also provide numerical results to verify the effectiveness of the proposed theory.

1. Introduction

In their pioneering work [1, 2], Pecora and Carroll have shown that chaotic systems can be synchronized by introducing appropriate coupling. The notion of synchronization of chaos has further been explored in secure communications of analog and digital signals [3] and for developing safe and reliable cryptographic systems [4]. For the synchronization of chaotic systems, a variety of approaches have been proposed which include nonlinear feedback [5], adaptive [6, 7], and active controls [8, 9].

Antisynchronization (AS) is a phenomenon in which the state vectors of the synchronized systems have the same amplitude but opposite signs to those of the driving system. Hence the sum of two signals converges to zero when AS appears. Antisynchronization has applications in lasers [10], in periodic oscillators, and in communication systems. Using AS to lasers, one may generate not only drop-outs of the intensity but also short pulses of high intensity, which results in the pulses of special shapes.

Active control method is used to AS for two identical integer order systems by Ho et al. [11] and for nonidentical systems by Li and Zhou [12]. Nonlinear control scheme was used by Li et al. [13] to study AS. Al-Sawalha [14] have reported AS between Chua's system and Nuclear spin generator (NSG) system. Recently AS between Lorenz system, Lü system, and Four-scroll system is investigated by Elabbasy and El-Dessoky [15].

Fractional calculus deals with derivatives and integration of arbitrary order [1618] and has deep and natural connections with many fields of applied mathematics, engineering, and physics. Fractional calculus has a wide range of applications in control theory [19], viscoelasticity [20], diffusion [2127], turbulence, electromagnetism, signal processing [28, 29], and bioengineering [30]. Analysis of fractional-order dynamical systems involving Riemann-Liouville as well as Caputo derivatives has been dealt with by present authors [31, 32].

Synchronization of fractional-order chaotic systems was first studied by Deng and Li [33] who carried out synchronization in case of the fractional Lü system. Further they have investigated synchronization of fractional Chen system [34]. Li and Deng have summarized the theory and techniques of synchronization in [35]. The theory for synchronization problems in an 𝜔-symmetrically coupled fractional differential systems have been studied by Zhou and Li [36]. Since then many fractional systems have been investigated by various researchers. A few examples in this regards are Li et al. [37] (Chua system), Wang et al. [38] (Chen system), Wang and Zhang [39] (unified system), Wang and He [40] (unified system), Yu and Li [41] (Rossler hyperchaos system), and Tavazoei and Haeri [42] (Lü system and Chen system). Of late Matouk [43] has synchronized fractional Lü system with fractional Chen system and fractional Chen system with fractional Lorenz system. Hu et al. [44] have synchronized fractional Lorenz and fractional Chen systems. Further Bhalekar and Daftardar-Gejji [45] have investigated the interrelationship between the (fractional) order and synchronization in different chaotic dynamical systems. However, it seems that there are no previous results on AS of two nonidentical fractional-order chaotic systems.

In the present paper, we study the antisynchronization of the following fractional systems using active control method: (i) Lorenz with Financial, (ii) Financial with Chen, and (iii) Lü with Financial.

2. Preliminaries

2.1. Fractional Calculus

Basic definitions and properties of fractional derivative/integrals are given below [16, 17, 46].

Definition 2.1. A real function 𝑓(𝑡),𝑡>0, is said to be in space 𝐶𝛼,𝛼 if there exists a real number 𝑝 (>𝛼), such that 𝑓(𝑡)=𝑡𝑝𝑓1(𝑡) where 𝑓1(𝑡)𝐶[0,).

Definition 2.2. A real function 𝑓(𝑡),𝑡>0, is said to be in space 𝐶𝑚𝛼,𝑚{0} if 𝑓(𝑚)𝐶𝛼.

Definition 2.3. Let 𝑓𝐶𝛼 and 𝛼1, then the (left-sided) Riemann-Liouville integral of order 𝜇,𝜇>0 is given by 𝐼𝜇1𝑓(𝑡)=Γ(𝜇)𝑡0(𝑡𝜏)𝜇1𝑓(𝜏)𝑑𝜏,𝑡>0.(2.1)

Definition 2.4. The (left-sided) Caputo fractional derivative of 𝑓, 𝑓𝐶𝑚1,𝑚{0}, is defined as 𝐷𝜇𝑑𝑓(𝑡)=𝑚𝑑𝑡𝑚𝑓(𝑡),𝜇=𝑚=𝐼𝑚𝜇𝑑𝑚𝑓(𝑡)𝑑𝑡𝑚,𝑚1<𝜇<𝑚,𝑚.(2.2) Note that for 𝑚1<𝜇𝑚,𝑚, 𝐼𝜇𝐷𝜇𝑓(𝑡)=𝑓(𝑡)𝑚1𝑘=0𝑑𝑘𝑓𝑑𝑡𝑘𝑡(0)𝑘,𝐼𝑘!𝜇𝑡𝜈=Γ(𝜈+1)𝑡Γ(𝜇+𝜈+1)𝜇+𝜈.(2.3)

2.2. Numerical Method for Solving Fractional Differential Equations

Numerical methods used for solving ODEs have to be modified for solving fractional differential equations (FDEs). A modification of Adams-Bashforth-Moulton algorithm is proposed by Diethelm et al. in [4749] to solve FDEs.

Consider for 𝛼(𝑚1,𝑚] the initial value problem (IVP)𝐷𝛼𝑦𝑦(𝑡)=𝑓(𝑡,𝑦(𝑡)),0𝑡𝑇,(𝑘)(0)=𝑦0(𝑘),𝑘=0,1,,𝑚1.(2.4) The IVP (2.4) is equivalent to the Volterra integral equation𝑦(𝑡)=𝑚1𝑘=0𝑦0(𝑘)𝑡𝑘+1𝑘!Γ(𝛼)𝑡0(𝑡𝜏)𝛼1𝑓(𝜏,𝑦(𝜏))𝑑𝜏.(2.5) Consider the uniform grid {𝑡𝑛=𝑛/𝑛=0,1,,𝑁} for some integer 𝑁 and =𝑇/𝑁. Let 𝑦(𝑡𝑛) be approximation to 𝑦(𝑡𝑛). Assume that we have already calculated approximations 𝑦(𝑡𝑗),𝑗=1,2,,𝑛, and we want to obtain 𝑦(𝑡𝑛+1) by means of the equation𝑦𝑡𝑛+1=𝑚1𝑘=0𝑡𝑘𝑛+1𝑦𝑘!0(𝑘)+𝛼Γ𝑓𝑡(𝛼+2)𝑛+1,𝑦𝑃𝑡𝑛+1+𝛼Γ(𝛼+2)𝑛𝑗=0𝑎𝑗,𝑛+1𝑓𝑡𝑗,𝑦𝑛𝑡𝑗,(2.6) where𝑎𝑗,𝑛+1=𝑛𝛼+1(𝑛𝛼)(𝑛+1)𝛼if𝑗=0,(𝑛𝑗+2)𝛼+1+(𝑛𝑗)𝛼+12(𝑛𝑗+1)𝛼+1if1𝑗𝑛,1if𝑗=𝑛+1.(2.7) The preliminary approximation 𝑦𝑃(𝑡𝑛+1) is called predictor and is given by𝑦𝑃𝑡𝑛+1=𝑚1𝑘=0𝑡𝑘𝑛+1𝑦𝑘!0(𝑘)+1Γ(𝛼)𝑛𝑗=0𝑏𝑗,𝑛+1𝑓𝑡𝑗,𝑦𝑛𝑡𝑗,(2.8) where𝑏𝑗,𝑛+1=𝛼𝛼((𝑛+1𝑗)𝛼(𝑛𝑗)𝛼).(2.9) Error in this method ismax𝑗=0,1,,𝑁||𝑦𝑡𝑗𝑦𝑡𝑗||=𝑂𝑝,(2.10) where 𝑝=min(2,1+𝛼).

3. System Description

The fractional-order Lorenz system [50, 51] is described by𝐷𝛼𝐷𝑥=𝜎(𝑦𝑥),𝛼𝐷𝑦=𝑟𝑥𝑦𝑥𝑧,𝛼𝑧=𝑥𝑦𝜇𝑧,(3.1) where 𝜎=10 is the Prandtl number, 𝑟=28 is the Rayleigh number over the critical Rayleigh number, and 𝜇=8/3 gives the size of the region approximated by the system. The minimum effective dimension for this system is 2.97 [51].

In [52] Chen proposed the financial system to fractional-order𝐷𝛼𝐷𝑥=𝑧+(𝑦𝑎)𝑥,𝛼𝑦=1𝑏𝑦𝑥2,𝐷𝛼𝑧=𝑥𝑐𝑧,(3.2) where 𝑎=3,𝑏=0.1, and 𝑐=1. The minimum effective dimension for which the system exhibits chaos is given by 2.32 [52].

Li and Peng [53] studied chaos in fractional-order Chen system𝐷𝛼𝑥=𝑎1(𝐷𝑦𝑥),𝛼𝑐𝑦=1𝑎1𝑥𝑥𝑧+𝑐1𝐷𝑦,𝛼𝑧=𝑥𝑦𝑏1𝑧,(3.3) where 𝑎1=35,𝑏1=3, and 𝑐1=27. The minimum effective dimension reported is 2.92 [53].

Fractional-order Lü system is the lowest-order chaotic system among all the chaotic systems reported in the literature [54]. The minimum effective dimension reported is 0.30. The system is given by𝐷𝛼𝑥=𝑎2(𝐷𝑦𝑥),𝛼𝑦=𝑐2𝐷𝑦𝑥𝑧,𝛼𝑧=𝑥𝑦𝑏2𝑧,(3.4) where 𝑎2=35,𝑏2=3, and 𝑐2=28.

4. Antisynchronization between Fractional-Order Lorenz and Financial System

In this section, we study the antisynchronization between Lorenz and Financial systems. Assuming that the Lorenz system drives the Financial system, we define the drive (master) and response (slave) systems as follows: 𝐷𝛼𝑥1𝑦=𝜎1𝑥1,𝐷𝛼𝑦1=𝑟𝑥1𝑦1𝑥1𝑧1,𝐷𝛼𝑧1=𝑥1𝑦1𝜇𝑧1,𝐷(4.1)𝛼𝑥2=𝑧2+𝑦2𝑥𝑎2+𝑢1𝐷(𝑡),𝛼𝑦2=1𝑏𝑦2𝑥22+𝑢2𝐷(𝑡),𝛼𝑧2=𝑥2𝑐𝑧2+𝑢3(𝑡).(4.2) The unknown terms 𝑢1,𝑢2,𝑢3 in (4.2) are active control functions to be determined. Define the error functions as𝑒1=𝑥1+𝑥2,𝑒2=𝑦1+𝑦2,𝑒3=𝑧1+𝑧2.(4.3) Equation (4.3) together with (4.1) and (4.2) yields the error system𝐷𝛼𝑒1=(𝑎𝜎)𝑥1+𝜎𝑦1+𝑥1𝑦1𝑧1𝑎𝑒1𝑦1𝑒1𝑥1𝑒2+𝑒1𝑒2+𝑒3+𝑢1(𝐷𝑡),𝛼𝑒2=1+𝑟𝑥1𝑥21+(𝑏1)𝑦1𝑥1𝑧1+2𝑥1𝑒1𝑒21𝑏𝑒2+𝑢2𝐷(𝑡),𝛼𝑒3=𝑥1+(𝑐𝜇)𝑧1+𝑥1𝑦1𝑒1𝑐𝑒3+𝑢3(𝑡).(4.4) We define active control functions 𝑢𝑖(𝑡) as𝑢1(𝑡)=𝑉1(𝑡)(𝑎𝜎)𝑥1𝜎𝑦1𝑥1𝑦1+𝑧1+𝑦1𝑒1+𝑥1𝑒2𝑒1𝑒2,𝑢2(𝑡)=𝑉2(𝑡)1𝑟𝑥1+𝑥21(𝑏1)𝑦1+𝑥1𝑧12𝑥1𝑒1+𝑒21,𝑢3(𝑡)=𝑉3(𝑡)𝑥1(𝑐𝜇)𝑧1𝑥1𝑦1.(4.5) The terms 𝑉𝑖(𝑡) are linear functions of the error terms 𝑒𝑖(𝑡). With the choice of 𝑢𝑖(𝑡) given by (4.5), the error system (4.5) becomes𝐷𝛼𝑒1=𝑎𝑒1𝑒3+𝑉1(𝐷𝑡),𝛼𝑒2=𝑏𝑒2+𝑉2𝐷(𝑡),𝛼𝑒3=𝑒1𝑐𝑒3+𝑉3(𝑡).(4.6) The control terms 𝑉𝑖(𝑡) are chosen so that the system (4.6) becomes stable. There is not a unique choice for such functions. We choose𝑉1𝑉2𝑉3𝑒=𝐴1e2𝑒3,(4.7) where 𝐴 is a 3×3 real matrix, chosen so that for all eigenvalues 𝜆𝑖 of the system (4.6) the condition||𝜆arg𝑖||>𝛼𝜋2(4.8) is satisfied. (The stability condition (4.8) is discussed in the literature [5557]). If we choose𝐴=𝑎10101+𝑏010𝑐1,(4.9) then the eigenvalues of the linear system (4.6) are 1,1, and −1. Hence the condition (4.8) is satisfied for 𝛼<2. Since we consider only the values 𝛼1, we get the required antisynchronization.

4.1. Simulation and Results

Parameters of the Lorenz system are taken as 𝜎=10,𝑟=28,𝜇=8/3 and Financial system as 𝑎=3,𝑏=0.1,𝑐=1. The fractional-order 𝛼 is taken to be 0.99 for which both the systems are chaotic. The initial conditions for drive and response system are 𝑥1(0)=10,𝑦1(0)=5,𝑧1(0)=10 and 𝑥2(0)=2,𝑦2(0)=3,𝑧2(0)=2, respectively. Initial conditions for the error system are thus 𝑒1(0)=12,𝑒2(0)=8, and 𝑒3(0)=12. Figures 1(a)1(c) show the antisynchronization between Lorenz and Financial system; the response system is given by dashed line. The errors 𝑒1(𝑡) (solid line), 𝑒2(𝑡) (dashed line) and 𝑒3(𝑡) (dot-dashed line) in the anti-synchronization are shown in Figure 1(d).

fig1
Figure 1: (a) Signals 𝑥1,𝑥2, (b) Signals 𝑦1,𝑦2, (c) Signals 𝑧1,𝑧2, and (d) Error system.

5. Antisynchronization between Financial and Chen Systems of Fractional Order

Assuming that Chen system is antisynchronized with Financial system; define the drive system as𝐷𝛼𝑥1=𝑧1+𝑦1𝑥𝑎1,𝐷𝛼𝑦1=1𝑏𝑦1𝑥21,𝐷𝛼𝑧1=𝑥1𝑐𝑧1(5.1) and the response system as𝐷𝛼𝑥2=𝑎1𝑦2𝑥2+𝑢4,𝐷𝛼𝑦2=𝑐1𝑎1𝑥2𝑥2𝑧2+𝑐1𝑦2+𝑢5,𝐷𝛼𝑧2=𝑥2𝑦2𝑏1𝑧2+𝑢6.(5.2) Let 𝑒1=𝑥1+𝑥2,𝑒2=𝑦1+𝑦2, and 𝑒3=𝑧1+𝑧2 be error functions. For antisynchronization, it is essential that the errors 𝑒𝑖0 as 𝑡. Note that𝐷𝛼𝑒1=𝑎1𝑥𝑎1𝑎1𝑦1+𝑥1𝑦1+𝑧1𝑎1𝑒1+𝑎1𝑒2+𝑢4𝐷(𝑡),𝛼𝑒2𝑎=1+1𝑐1𝑥1𝑥21𝑏+𝑐1𝑦1𝑥1𝑧1+𝑐1𝑎1𝑒1+𝑧1𝑒1+𝑐1𝑒2+𝑥1𝑒3𝑒1𝑒3+𝑢5𝐷(𝑡),𝛼𝑒3=𝑥1+𝑥1𝑦1+𝑏1𝑧𝑐1𝑦1𝑒1𝑥1𝑒2+𝑒1𝑒2𝑏1𝑒3+𝑢6(𝑡).(5.3) The control functions are chosen as𝑢4=𝑉4𝑎1𝑥𝑎1+𝑎1𝑦1𝑥1𝑦1𝑧1,𝑢5=𝑉5𝑎11𝑐1𝑥1+𝑥21+𝑏+𝑐1𝑦1+𝑥1𝑧1𝑧1𝑒1𝑥1𝑒3+𝑒1𝑒3,𝑢6=𝑉6+𝑥1𝑥1𝑦1𝑏1𝑧c1+𝑦1𝑒1+𝑥1𝑒2𝑒1𝑒2.(5.4) The linear functions 𝑉4,𝑉5,𝑉6 are given by𝑉4=𝑎1𝑒11𝑎1𝑒2,𝑉5𝑎=1𝑐1𝑒1𝑐1𝑒+12,𝑉6=𝑏1𝑒13.(5.5) With the values given in (5.4) and (5.5), the error system (5.3) becomes𝐷𝛼𝑒1𝐷𝛼𝑒2𝐷𝛼𝑒3=𝑒1000100011𝑒2𝑒3.(5.6) It can be observed that the coefficient matrix of the error system (5.6) has eigenvalues 1,1,1. So the system is stable and antisynchronization is achieved.

5.1. Simulations and Results

We take parameters for fractional-order Chen system as 𝑎1=35,𝑏1=3,𝑐1=27. Parameters for the Financial system are same as given in Section 4.1. Experiments are done for fixed value of fractional-order 𝛼=0.95, which is same for drive and response system (5.1) and (5.2). The initial conditions for the systems (5.1) and (5.2) are 𝑥1(0)=2,𝑦1(0)=3,𝑧1(0)=2 and 𝑥2(0)=10,𝑦2(0)=25,𝑧2(0)=36, respectively. For the error system (5.6), the initial conditions turns out to be 𝑒1(0)=12,𝑒2(0)=28,𝑒3(0)=38. The simulation results are summarized in Figure 2. Antisynchronization between fractional Financial and Chen system is shown in Figure 2(a) (signals 𝑥1,𝑥2), Figure 2(b) (signals 𝑦1,𝑦2), and Figure 2(c) (signals 𝑧1,𝑧2). Note that the drive systems are shown by solid lines, whereas response systems are shown by dashed lines. The errors 𝑒1(𝑡) (solid line), 𝑒2(𝑡) (dashed line), and 𝑒3(𝑡) (dot-dashed line) in the antisynchronization are shown in Figure 2(d).

fig2
Figure 2: (a) Signals 𝑥1,𝑥2, (b) Signals 𝑦1,𝑦2, (c) Signals 𝑧1,𝑧2, and (d) Error system.

6. Antisynchronization between Fractional Lü and Financial System

In this case, consider Lü system as the drive system𝐷𝛼𝑥1=𝑎2𝑦1𝑥1,𝐷𝛼𝑦1=𝑐2𝑦1𝑥1𝑧1,𝐷𝛼𝑧1=𝑥1𝑦1𝑏2𝑧1,(6.1) and the response system as the Financial system𝐷𝛼𝑥2=𝑧2+𝑦2𝑥𝑎2+𝑢7,𝐷𝛼𝑦2=1𝑏𝑦2𝑥22+𝑢8,𝐷𝛼𝑧2=𝑥2𝑐𝑧2+𝑢9.(6.2)

Let 𝑒1=𝑥1+𝑥2,𝑒2=𝑦1+𝑦2, and 𝑒3=𝑧1+𝑧2 be error functions. For antisynchronization, it is essential that the errors 𝑒𝑖0 as 𝑡. To achieve this one should choose the control terms 𝑢7,𝑢8,𝑢9 properly. The error system thus becomes𝐷𝛼𝑒1=𝑎𝑎2𝑥1+𝑎2𝑦1+𝑥1𝑦1𝑧1𝑎𝑒1𝑦1𝑒1𝑥1𝑒2+𝑒1𝑒2+𝑒3+𝑢7,𝐷𝛼𝑒2=1𝑥21+𝑏+𝑐2𝑦1𝑥1𝑧1+2𝑥1𝑒1𝑒21𝑏𝑒2+𝑢8,𝐷𝛼𝑒3=𝑥1+𝑥1𝑦1+𝑐𝑏2𝑧1𝑒1𝑐𝑒3+𝑢9.(6.3) The control functions are chosen as𝑢7=𝑉7𝑎𝑎2𝑥1𝑎2𝑦1𝑥1𝑦1+𝑧1+𝑦1𝑒1+𝑥1𝑒2𝑒1𝑒2,𝑢8=𝑉81+𝑥21𝑏+𝑐2𝑦1+𝑥1𝑧12𝑥1𝑒1+𝑒21,𝑢9=𝑉9𝑥1𝑥1𝑦1𝑐𝑏2𝑧1.(6.4) The linear functions 𝑉7,𝑉8,𝑉9 are given by𝑉7=(𝑎1)𝑒1𝑒3,𝑉8=(1+𝑏)𝑒2,𝑉9=𝑒1+(𝑐1)𝑒3.(6.5) With the values given in (6.4) and (6.5), the error system (6.3) becomes𝐷𝛼𝑒1𝐷𝛼𝑒2𝐷𝛼𝑒3=𝑒1000100011𝑒2𝑒3.(6.6) It can be observed that the coefficient matrix of the error system (6.6) has eigenvalues 1,1,1. So the system is stable and antisynchronization is achieved.

6.1. Simulations and Results

Parameters for the Lü system are 𝑎2=35,𝑏2=3,𝑐2=28, whereas parameters for Financial system are unaltered. The initial conditions for drive system are 𝑥1(0)=0.2,𝑦1(0)=0,𝑧1(0)=0.5, whereas the initial conditions for response system are 𝑥2(0)=2,𝑦2(0)=3,𝑧2(0)=2. Hence the initial conditions for the error system (6.6) are 𝑒1(0)=2.2,𝑒2(0)=3,𝑒3(0)=2.5. We perform the numerical simulations for fractional order 𝛼, namely, 0.91 of the drive system (6.1) and response system (6.2). Figures 3(a), 3(b), and 3(c) show antisynchronization between fractional Lü and Financial system for 𝛼=0.91. Figure 3(d) shows the errors 𝑒1(𝑡) (solid line), 𝑒2(𝑡) (dashed line), and 𝑒3(𝑡) (dot-dashed line) in the antisynchronization for 𝛼=0.91.

fig3
Figure 3: (a) 𝛼=0.91, Signals 𝑥1,𝑥2, (b) 𝛼=0.91, Signals 𝑦1,𝑦2, (c) 𝛼=0.91, Signals 𝑧1,𝑧2, and (d) 𝛼=0.91, Error system.

Mathematica 7 has been used for computations in the present paper.

7. Conclusions

Antisynchronization of nonidentical fractional-order chaotic systems has been done first time in the literature using active control. The fractional Financial system is controlled by fractional Lorenz system, the fractional Chen system is controlled by fractional Financial system, and the fractional Financial system is controlled by fractional Lü system.

Acknowledgment

V. Daftardar-Gejji acknowledges the Department of Science and Technology, N. Delhi, India for the Research Grants (project no. SR/S2/HEP-024/2009).

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