Antisynchronization of Nonidentical Fractional-Order Chaotic Systems Using Active Control

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 [16–18] 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 [21–27], 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 𝐶_𝛼,𝛼∈ℜ if there exists a real number 𝑝 (>𝛼), such that 𝑓(𝑡)=𝑡^𝑝𝑓₁(𝑡) where 𝑓₁(𝑡)∈𝐶[0,∞).

Definition 2.2. A real function 𝑓(𝑡),𝑡>0, is said to be in space 𝐶^𝑚𝛼,𝑚∈ℕ∪{0} if 𝑓^(𝑚)∈𝐶_𝛼.

Definition 2.3. Let 𝑓∈𝐶_𝛼 and 𝛼≥−1, then the (left-sided) Riemann-Liouville integral of order 𝜇,𝜇>0 is given by 𝐼^𝜇𝑓(𝑡)=1Γ(𝜇)^𝑡0(𝑡−𝜏)^𝜇−1𝑓(𝜏)𝑑𝜏,𝑡>0.(2.1)

Definition 2.4. The (left-sided) Caputo fractional derivative of 𝑓, 𝑓∈𝐶^𝑚−1,𝑚∈ℕ∪{0}, is defined as 𝐷^𝜇𝑓(𝑡)=𝑑^𝑚𝑑𝑡^𝑚=𝐼_𝑚−𝜇𝑓(𝑡),𝜇=𝑚𝑑^𝑚𝑓(𝑡)𝑑𝑡^𝑚,𝑚−1<𝜇<𝑚,𝑚∈ℕ.(2.2) Note that for 𝑚−1<𝜇≤𝑚,𝑚∈ℕ, 𝐼^𝜇𝐷^𝜇𝑓(𝑡)=𝑓(𝑡)−^𝑚−1_𝑘=0𝑑^𝑘𝑓𝑑𝑡^𝑘(0)𝑡^𝑘𝐼𝑘!_𝜇,𝑡^𝜈=Γ(𝜈+1)Γ(𝜇+𝜈+1)𝑡^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 [47–49] to solve FDEs.

Consider for 𝛼∈(𝑚−1,𝑚] the initial value problem (IVP)𝑦𝐷^𝛼𝑦(𝑡)=𝑓(𝑡,𝑦(𝑡)),0≤𝑡≤𝑇,_(𝑘)(0)=𝑦^0(𝑘),𝑘=0,1,…,𝑚−1.(2.4) The IVP (2.4) is equivalent to the Volterra integral equation𝑦(𝑡)=^𝑚−1_𝑘=0𝑦^0(𝑘)𝑡^𝑘𝑘!+Γ1(𝛼)^𝑡0(𝑡−𝜏)^𝛼−1𝑓(𝜏,𝑦(𝜏))𝑑𝜏.(2.5) Consider the uniform grid {𝑡_𝑛=𝑛ℎ/𝑛=0,1,…,𝑁} for some integer 𝑁 and ℎ∶=𝑇/𝑁. Let 𝑦_ℎ(𝑡_𝑛) be approximation to 𝑦(𝑡_𝑛). Assume that we have already calculated approximations 𝑦_ℎ(𝑡_𝑗),𝑗=1,2,…,𝑛, and we want to obtain 𝑦_ℎ(𝑡_𝑛+1) by means of the equation𝑦_ℎ𝑡_𝑛+1=^𝑚−1_𝑘=0𝑡^𝑘𝑛+1𝑘!𝑦^0(𝑘)+ℎ^𝛼Γ(𝛼+2)𝑓𝑡^𝑛+1,𝑦^𝑃ℎ𝑡_𝑛+1+ℎ^𝛼Γ(𝛼+2)^𝑛_𝑗=0𝑎_𝑗,𝑛+1𝑓𝑡_𝑗,𝑦_𝑛𝑡_𝑗,(2.6) where𝑎_𝑗,𝑛+1=⎧⎪⎨⎪⎩𝑛^𝛼+1(𝑛−𝑗−+(𝑛2−)𝛼_𝛼+1)(𝑛+1)^𝛼if𝑗=0,1+(𝑛−𝑗)^𝛼+1−2(𝑛−𝑗+1)^𝛼+1iiff1𝑗=≤𝑛𝑗+≤1𝑛.,(2.7) The preliminary approximation 𝑦^𝑃ℎ(𝑡_𝑛+1) is called predictor and is given by𝑦^𝑃ℎ𝑡_𝑛+1=^𝑚−1_𝑘=0𝑡^𝑘𝑛+1𝑘!𝑦^0(𝑘)+1Γ(𝛼)^𝑛_𝑗=0𝑏_𝑗,𝑛+1𝑓𝑡_𝑗,𝑦_𝑛𝑡_𝑗,(2.8) where𝑏_𝑗,𝑛+1=ℎ^𝛼𝛼((𝑛+1−𝑗)^𝛼−(𝑛−𝑗)^𝛼).(2.9) Error in this method ismax_{𝑗=0,1,…,𝑁}||𝑦𝑡_𝑗−𝑦_ℎ𝑡_𝑗||=𝑂ℎ^𝑝,(2.10) where 𝑝=min(2,1+𝛼).

3. System Description

The fractional-order Lorenz system [50, 51] is described by𝐷𝐷^𝛼𝑥=𝜎(𝑦−𝑥),_𝛼𝐷𝑦=𝑟𝑥−𝑦−𝑥𝑧,_𝛼𝑧=𝑥𝑦−𝜇𝑧,(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-order𝐷𝐷^𝛼𝑥=𝑧+(𝑦−𝑎)𝑥,_𝛼𝐷_𝛼𝑦=1−𝑏𝑦−𝑥²,𝑧=−𝑥−𝑐𝑧,(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 system𝐷𝐷^𝛼𝑥=𝑎₁(𝑦−𝑥),_𝛼𝐷𝑦=𝑐₁−𝑎₁𝑥−𝑥𝑧+𝑐₁𝑦,_𝛼𝑧=𝑥𝑦−𝑏₁𝑧,(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 by𝐷𝐷^𝛼𝑥=𝑎₂(𝑦−𝑥),_𝛼𝐷𝑦=𝑐₂𝑦−𝑥𝑧,_𝛼𝑧=𝑥𝑦−𝑏₂𝑧,(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: 𝐷𝐷^𝛼_𝛼𝑥₁=𝜎𝑦₁−𝑥₁,𝐷_𝛼𝑦₁=𝑟𝑥₁−𝑦₁−𝑥₁𝑧₁,𝐷𝑧₁=𝑥₁𝑦₁−𝜇𝑧₁,(4.1)_𝛼𝐷𝑥₂=𝑧₂+𝑦₂−𝑎𝑥₂+𝑢₁(𝑡),_𝛼𝐷𝑦₂=1−𝑏𝑦₂−𝑥²²+𝑢₂(𝑡),_𝛼𝑧₂=−𝑥₂−𝑐𝑧₂+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 system𝐷𝐷^𝛼𝑒₁=(𝑎−𝜎)𝑥₁+𝜎𝑦₁+𝑥₁𝑦₁−𝑧₁−𝑎𝑒₁−𝑦₁𝑒₁−𝑥₁𝑒₂+𝑒₁𝑒₂+𝑒₃+𝑢₁(𝑡),_𝛼𝐷𝑒₂=1+𝑟𝑥₁−𝑥²¹+(𝑏−1)𝑦₁−𝑥₁𝑧₁+2𝑥₁𝑒₁−𝑒²¹−𝑏𝑒₂+𝑢₂(𝑡),_𝛼𝑒₃=𝑥₁+(𝑐−𝜇)𝑧₁+𝑥₁𝑦₁−𝑒₁−𝑐𝑒₃+𝑢₃(𝑡).(4.4) We define active control functions 𝑢_𝑖(𝑡) as𝑢𝑢₁₂(𝑡)=𝑉₁(𝑡)−(𝑎−𝜎)𝑥₁−𝜎𝑦₁−𝑥₁𝑦₁+𝑧₁+𝑦₁𝑒₁+𝑥₁𝑒₂−𝑒₁𝑒₂,𝑢₃(𝑡)=𝑉₂(𝑡)−1−𝑟𝑥₁+𝑥²¹−(𝑏−1)𝑦₁+𝑥₁𝑧₁−2𝑥₁𝑒₁+𝑒²¹,(𝑡)=𝑉₃(𝑡)−𝑥₁−(𝑐−𝜇)𝑧₁−𝑥₁𝑦₁.(4.5) The terms 𝑉_𝑖(𝑡) are linear functions of the error terms 𝑒_𝑖(𝑡). With the choice of 𝑢_𝑖(𝑡) given by (4.5), the error system (4.5) becomes𝐷𝐷^𝛼𝑒₁=−𝑎𝑒₁−𝑒₃+𝑉₁(𝑡),_𝛼𝐷𝑒₂=−𝑏𝑒₂+𝑉₂(𝑡),_𝛼𝑒₃=−𝑒₁−𝑐𝑒₃+𝑉₃(𝑡).(4.6) The control terms 𝑉_𝑖(𝑡) are chosen so that the system (4.6) becomes stable. There is not a unique choice for such functions. We choose⎛⎜⎜⎜⎜⎝𝑉₁𝑉₂𝑉₃⎞⎟⎟⎟⎟⎠=𝐴⎛⎜⎜⎜⎜⎝𝑒₁e₂𝑒₃⎞⎟⎟⎟⎟⎠,(4.7) where 𝐴 is a 3×3 real matrix, chosen so that for all eigenvalues 𝜆_𝑖 of the system (4.6) the condition||arg𝜆_𝑖||>𝛼𝜋2(4.8) is satisfied. (The stability condition (4.8) is discussed in the literature [55–57]). If we choose𝐴=⎛⎜⎜⎜⎜⎝𝑎−011−100+𝑏𝑐−−011⎞⎟⎟⎟⎟⎠,(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 𝛼≤1, 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).

(a)

(b)

(c)

(d)

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 as𝐷𝐷^𝛼_𝛼𝑥₁=𝑧₁+𝑦₁−𝑎𝑥₁,𝐷_𝛼𝑦₁=1−𝑏𝑦₁−𝑥²¹,𝑧₁=−𝑥₁−𝑐𝑧₁(5.1) and the response system as𝐷𝐷^𝛼_𝛼𝑥₂=𝑎₁𝑦₂−𝑥₂+𝑢₄,𝐷_𝛼𝑦₂=𝑐₁−𝑎₁𝑥₂−𝑥₂𝑧₂+𝑐₁𝑦₂+𝑢₅,𝑧₂=𝑥₂𝑦₂−𝑏₁𝑧₂+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 𝑒_𝑖→0 as 𝑡→∞. Note that𝐷𝐷^𝛼𝑒₁=𝑎₁−𝑎𝑥₁−𝑎₁𝑦₁+𝑥₁𝑦₁+𝑧₁−𝑎₁𝑒₁+𝑎₁𝑒₂+𝑢₄(𝑡),_𝛼𝑒₂=1+𝑎₁−𝑐₁𝑥₁−𝑥²¹−𝑏+𝑐₁𝑦₁−𝑥₁𝑧₁𝐷+𝑐₁−𝑎₁𝑒₁+𝑧₁𝑒₁+𝑐₁𝑒₂+𝑥₁𝑒₃−𝑒₁𝑒₃+𝑢₅(𝑡),_𝛼𝑒₃=−𝑥₁+𝑥₁𝑦₁+𝑏₁−𝑐𝑧₁−𝑦₁𝑒₁−𝑥₁𝑒₂+𝑒₁𝑒₂−𝑏₁𝑒₃+𝑢₆(𝑡).(5.3) The control functions are chosen as𝑢𝑢₄₅=𝑉₄−𝑎₁−𝑎𝑥₁+𝑎₁𝑦₁−𝑥₁𝑦₁−𝑧₁,𝑢₆=𝑉₅−1−𝑎₁−𝑐₁𝑥₁+𝑥²¹+𝑏+𝑐₁𝑦₁+𝑥₁𝑧₁−𝑧₁𝑒₁−𝑥₁𝑒₃+𝑒₁𝑒₃,=𝑉₆+𝑥₁−𝑥₁𝑦₁−𝑏₁−c𝑧₁+𝑦₁𝑒₁+𝑥₁𝑒₂−𝑒₁𝑒₂.(5.4) The linear functions 𝑉₄,𝑉₅,𝑉₆ are given by𝑉𝑉₄₅=𝑎₁−1𝑒₁−𝑎₁𝑒₂,𝑉₆=−𝑎₁−𝑐₁𝑒₁−𝑐₁+1𝑒₂,=𝑏₁−1𝑒₃.(5.5) With the values given in (5.4) and (5.5), the error system (5.3) becomes⎛⎜⎜⎜⎜⎝𝐷^𝛼𝑒₁𝐷^𝛼𝑒₂𝐷^𝛼𝑒₃⎞⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎝−001−001−001⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝𝑒₁𝑒₂𝑒₃⎞⎟⎟⎟⎟⎠.(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).

(a)

(b)

(c)

(d)

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 system𝐷𝐷^𝛼_𝛼𝑥₁=𝑎₂𝑦₁−𝑥₁,𝐷_𝛼𝑦₁=𝑐₂𝑦₁−𝑥₁𝑧₁,𝑧₁=𝑥₁𝑦₁−𝑏₂𝑧₁,(6.1) and the response system as the Financial system𝐷𝐷^𝛼_𝛼𝑥₂=𝑧₂+𝑦₂−𝑎𝑥₂+𝑢₇,𝐷_𝛼𝑦₂=1−𝑏𝑦₂−𝑥²²+𝑢₈,𝑧₂=−𝑥₂−𝑐𝑧₂+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 𝑒_𝑖→0 as 𝑡→∞. To achieve this one should choose the control terms 𝑢₇,𝑢₈,𝑢₉ properly. The error system thus becomes𝐷𝐷^𝛼_𝛼𝑒₁=𝑎−𝑎₂𝑥₁+𝑎₂𝑦₁+𝑥₁𝑦₁−𝑧₁−𝑎𝑒₁−𝑦₁𝑒₁−𝑥₁𝑒₂+𝑒₁𝑒₂+𝑒₃+𝑢₇,𝐷_𝛼𝑒₂=1−𝑥²¹+𝑏+𝑐₂𝑦₁−𝑥₁𝑧₁+2𝑥₁𝑒₁−𝑒²¹−𝑏𝑒₂+𝑢₈,𝑒₃=𝑥₁+𝑥₁𝑦₁+𝑐−𝑏₂𝑧₁−𝑒₁−𝑐𝑒₃+𝑢₉.(6.3) The control functions are chosen as𝑢𝑢₇₈=𝑉₇−𝑎−𝑎₂𝑥₁−𝑎₂𝑦₁−𝑥₁𝑦₁+𝑧₁+𝑦₁𝑒₁+𝑥₁𝑒₂−𝑒₁𝑒₂,𝑢₉=𝑉₈−1+𝑥²¹−𝑏+𝑐₂𝑦₁+𝑥₁𝑧₁−2𝑥₁𝑒₁+𝑒²¹,=𝑉₉−𝑥₁−𝑥₁𝑦₁−𝑐−𝑏₂𝑧₁.(6.4) The linear functions 𝑉₇,𝑉₈,𝑉₉ are given by𝑉𝑉₇₈=(𝑎−1)𝑒₁−𝑒₃,𝑉₉=(−1+𝑏)𝑒₂,=𝑒₁+(𝑐−1)𝑒₃.(6.5) With the values given in (6.4) and (6.5), the error system (6.3) becomes⎛⎜⎜⎜⎜⎝𝐷^𝛼𝑒₁𝐷^𝛼𝑒₂𝐷^𝛼𝑒₃⎞⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎝−001−001−001⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝𝑒₁𝑒₂𝑒₃⎞⎟⎟⎟⎟⎠.(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.

(a)

(b)

(c)

(d)

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).