# Antisynchronization of  Nonidentical Fractional-Order Chaotic Systems Using Active Control

Authors: Sachin Bhalekar; Varsha Daftardar-Gejji
Journal: International Journal of Differential Equations
 (2011)

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## 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.


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## Body

## 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 functionf(t),t>0, is said to be in space Cα,α∈ℜ if there exists a real number p (>α), such that f(t)=tpf1(t) where f1(t)∈C[0,∞).Definition 2.2.
A real functionf(t),t>0, is said to be in space Cαm,m∈ℕ∪{0} if f(m)∈Cα.Definition 2.3.
Letf∈Cα and α≥-1, then the (left-sided) Riemann-Liouville integral of order μ,μ>0 is given by
(2.1)Iμf(t)=1Γ(μ)∫0t(t-τ)μ-1f(τ)dτ,t>0.Definition 2.4.
The (left-sided) Caputo fractional derivative off, f∈C-1m,m∈ℕ∪{0}, is defined as
(2.2)Dμf(t)=dmdtmf(t),μ=m=Im-μdmf(t)dtm,m-1<μ<m,m∈N.
Note that for m-1<μ≤m,m∈ℕ,
(2.3)IμDμf(t)=f(t)-∑k=0m-1dkfdtk(0)tkk!,Iμtν=Γ(ν+1)Γ(μ+ν+1)tμ+ν.


### 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 forα∈(m-1,m] the initial value problem (IVP)(2.4)Dαy(t)=f(t,y(t)),0≤t≤T,y(k)(0)=y0(k),k=0,1,…,m-1.
The IVP (2.4) is equivalent to the Volterra integral equation(2.5)y(t)=∑k=0m-1y0(k)tkk!+1Γ(α)∫0t(t-τ)α-1f(τ,y(τ))dτ.
Consider the uniform grid {tn=nh/n=0,1,…,N} for some integer N and h:=T/N. Let yh(tn) be approximation to y(tn). Assume that we have already calculated approximations yh(tj),j=1,2,…,n, and we want to obtain yh(tn+1) by means of the equation(2.6)yh(tn+1)=∑k=0m-1tn+1kk!y0(k)+hαΓ(α+2)f(tn+1,yhP(tn+1))+hαΓ(α+2)∑j=0naj,n+1f(tj,yn(tj)),
where(2.7)aj,n+1={nα+1-(n-α)(n+1)αifj=0,(n-j+2)α+1+(n-j)α+1-2(n-j+1)α+1if1≤j≤n,1ifj=n+1.
The preliminary approximation yhP(tn+1) is called predictor and is given by(2.8)yhP(tn+1)=∑k=0m-1tn+1kk!y0(k)+1Γ(α)∑j=0nbj,n+1f(tj,yn(tj)),
where(2.9)bj,n+1=hαα((n+1-j)α-(n-j)α).
Error in this method is(2.10)maxj=0,1,…,N|y(tj)-yh(tj)|=O(hp),
where p=min(2,1+α).


## 2.1. Fractional Calculus

Basic definitions and properties of fractional derivative/integrals are given below [16, 17, 46].Definition 2.1.
A real functionf(t),t>0, is said to be in space Cα,α∈ℜ if there exists a real number p (>α), such that f(t)=tpf1(t) where f1(t)∈C[0,∞).Definition 2.2.
A real functionf(t),t>0, is said to be in space Cαm,m∈ℕ∪{0} if f(m)∈Cα.Definition 2.3.
Letf∈Cα and α≥-1, then the (left-sided) Riemann-Liouville integral of order μ,μ>0 is given by
(2.1)Iμf(t)=1Γ(μ)∫0t(t-τ)μ-1f(τ)dτ,t>0.Definition 2.4.
The (left-sided) Caputo fractional derivative off, f∈C-1m,m∈ℕ∪{0}, is defined as
(2.2)Dμf(t)=dmdtmf(t),μ=m=Im-μdmf(t)dtm,m-1<μ<m,m∈N.
Note that for m-1<μ≤m,m∈ℕ,
(2.3)IμDμf(t)=f(t)-∑k=0m-1dkfdtk(0)tkk!,Iμtν=Γ(ν+1)Γ(μ+ν+1)tμ+ν.


## 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 forα∈(m-1,m] the initial value problem (IVP)(2.4)Dαy(t)=f(t,y(t)),0≤t≤T,y(k)(0)=y0(k),k=0,1,…,m-1.
The IVP (2.4) is equivalent to the Volterra integral equation(2.5)y(t)=∑k=0m-1y0(k)tkk!+1Γ(α)∫0t(t-τ)α-1f(τ,y(τ))dτ.
Consider the uniform grid {tn=nh/n=0,1,…,N} for some integer N and h:=T/N. Let yh(tn) be approximation to y(tn). Assume that we have already calculated approximations yh(tj),j=1,2,…,n, and we want to obtain yh(tn+1) by means of the equation(2.6)yh(tn+1)=∑k=0m-1tn+1kk!y0(k)+hαΓ(α+2)f(tn+1,yhP(tn+1))+hαΓ(α+2)∑j=0naj,n+1f(tj,yn(tj)),
where(2.7)aj,n+1={nα+1-(n-α)(n+1)αifj=0,(n-j+2)α+1+(n-j)α+1-2(n-j+1)α+1if1≤j≤n,1ifj=n+1.
The preliminary approximation yhP(tn+1) is called predictor and is given by(2.8)yhP(tn+1)=∑k=0m-1tn+1kk!y0(k)+1Γ(α)∑j=0nbj,n+1f(tj,yn(tj)),
where(2.9)bj,n+1=hαα((n+1-j)α-(n-j)α).
Error in this method is(2.10)maxj=0,1,…,N|y(tj)-yh(tj)|=O(hp),
where p=min(2,1+α).


## 3. System Description

The fractional-order Lorenz system [50, 51] is described by(3.1)Dαx=σ(y-x),Dαy=rx-y-xz,Dαz=xy-μz,
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-order(3.2)Dαx=z+(y-a)x,Dαy=1-by-x2,Dαz=-x-cz,
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 system(3.3)Dαx=a1(y-x),Dαy=(c1-a1)x-xz+c1y,Dαz=xy-b1z,
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 by(3.4)Dαx=a2(y-x),Dαy=c2y-xz,Dαz=xy-b2z,
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:(4.1)Dαx1=σ(y1-x1),Dαy1=rx1-y1-x1z1,Dαz1=x1y1-μz1,(4.2)Dαx2=z2+(y2-a)x2+u1(t),Dαy2=1-by2-x22+u2(t),Dαz2=-x2-cz2+u3(t).
The unknown terms u1,u2,u3 in (4.2) are active control functions to be determined. Define the error functions as(4.3)e1=x1+x2,e2=y1+y2,e3=z1+z2.
Equation (4.3) together with (4.1) and (4.2) yields the error system(4.4)Dαe1=(a-σ)x1+σy1+x1y1-z1-ae1-y1e1-x1e2+e1e2+e3+u1(t),Dαe2=1+rx1-x12+(b-1)y1-x1z1+2x1e1-e12-be2+u2(t),Dαe3=x1+(c-μ)z1+x1y1-e1-ce3+u3(t).
We define active control functions ui(t) as(4.5)u1(t)=V1(t)-(a-σ)x1-σy1-x1y1+z1+y1e1+x1e2-e1e2,u2(t)=V2(t)-1-rx1+x12-(b-1)y1+x1z1-2x1e1+e12,u3(t)=V3(t)-x1-(c-μ)z1-x1y1.
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) becomes(4.6)Dαe1=-ae1-e3+V1(t),Dαe2=-be2+V2(t),Dαe3=-e1-ce3+V3(t).
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 choose(4.7)(V1V2V3)=A(e1e2e3),
where A is a 3×3 real matrix, chosen so that for all eigenvalues λi of the system (4.6) the condition(4.8)|arg(λi)|>απ2
is satisfied. (The stability condition (4.8) is discussed in the literature [55–57]). If we choose(4.9)A=(a-10-10-1+b010c-1),
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 asσ=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) Signalsx1,x2, (b) Signals y1,y2, (c) Signals z1,z2, and (d) Error system.
(a)(b)(c)(d)


## 4.1. Simulation and Results

Parameters of the Lorenz system are taken asσ=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) Signalsx1,x2, (b) Signals y1,y2, (c) Signals z1,z2, and (d) Error system.
(a)(b)(c)(d)


## 5. Antisynchronization between Financial and Chen Systems of Fractional Order

Assuming that Chen system is antisynchronized with Financial system; define the drive system as(5.1)Dαx1=z1+(y1-a)x1,Dαy1=1-by1-x12,Dαz1=-x1-cz1
and the response system as(5.2)Dαx2=a1(y2-x2)+u4,Dαy2=(c1-a1)x2-x2z2+c1y2+u5,Dαz2=x2y2-b1z2+u6.
Let e1=x1+x2,e2=y1+y2, and e3=z1+z2 be error functions. For antisynchronization, it is essential that the errors ei→0 as t→∞. Note that(5.3)Dαe1=(a1-a)x1-a1y1+x1y1+z1-a1e1+a1e2+u4(t),Dαe2=1+(a1-c1)x1-x12-(b+c1)y1-x1z1+(c1-a1)e1+z1e1+c1e2+x1e3-e1e3+u5(t),Dαe3=-x1+x1y1+(b1-c)z1-y1e1-x1e2+e1e2-b1e3+u6(t).
The control functions are chosen as(5.4)u4=V4-(a1-a)x1+a1y1-x1y1-z1,u5=V5-1-(a1-c1)x1+x12+(b+c1)y1+x1z1-z1e1-x1e3+e1e3,u6=V6+x1-x1y1-(b1-c)z1+y1e1+x1e2-e1e2.
The linear functions V4,V5,V6 are given by(5.5)V4=(a1-1)e1-a1e2,V5=-(a1-c1)e1-(c1+1)e2,V6=(b1-1)e3.
With the values given in (5.4) and (5.5), the error system (5.3) becomes(5.6)(Dαe1Dαe2Dαe3)=(-1000-1000-1)(e1e2e3).
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 asa1=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) Signalsx1,x2, (b) Signals y1,y2, (c) Signals z1,z2, and (d) Error system.
(a)(b)(c)(d)


## 5.1. Simulations and Results

We take parameters for fractional-order Chen system asa1=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) Signalsx1,x2, (b) Signals y1,y2, (c) Signals z1,z2, and (d) Error system.
(a)(b)(c)(d)


## 6. Antisynchronization between Fractional Lü and Financial System

In this case, consider Lü system as the drive system(6.1)Dαx1=a2(y1-x1),Dαy1=c2y1-x1z1,Dαz1=x1y1-b2z1,
and the response system as the Financial system(6.2)Dαx2=z2+(y2-a)x2+u7,Dαy2=1-by2-x22+u8,Dαz2=-x2-cz2+u9.Lete1=x1+x2,e2=y1+y2, and e3=z1+z2 be error functions. For antisynchronization, it is essential that the errors ei→0 as t→∞. To achieve this one should choose the control terms u7,u8,u9 properly. The error system thus becomes(6.3)Dαe1=(a-a2)x1+a2y1+x1y1-z1-ae1-y1e1-x1e2+e1e2+e3+u7,Dαe2=1-x12+(b+c2)y1-x1z1+2x1e1-e12-be2+u8,Dαe3=x1+x1y1+(c-b2)z1-e1-ce3+u9.
The control functions are chosen as(6.4)u7=V7-(a-a2)x1-a2y1-x1y1+z1+y1e1+x1e2-e1e2,u8=V8-1+x12-(b+c2)y1+x1z1-2x1e1+e12,u9=V9-x1-x1y1-(c-b2)z1.
The linear functions V7,V8,V9 are given by(6.5)V7=(a-1)e1-e3,V8=(-1+b)e2,V9=e1+(c-1)e3.
With the values given in (6.4) and (6.5), the error system (6.3) becomes(6.6)(Dαe1Dαe2Dαe3)=(-1000-1000-1)(e1e2e3).
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 area2=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)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.
(a)(b)(c)(d)Mathematica 7 has been used for computations in the present paper.


## 6.1. Simulations and Results

Parameters for the Lü system area2=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)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.
(a)(b)(c)(d)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.



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