# Synchronization of Chaotic  Fractional-Order WINDMI Systems via  Linear State Error Feedback Control

Authors: Baogui Xin; Tong Chen; Yanqin Liu
Journal: Mathematical Problems in Engineering
 (2010)

---

## Abstract

We propose a fractional-order WINDMI system, as a generalization of an integer-order
system developed by Sprott (2003). The considered synchronization scheme consists of identical master and slave
fractional-order WINDMI systems coupled by linear state error variables. Based on the stability
theory of nonlinear fractional-order systems, linear state error feedback control technique is
applied to achieve chaos synchronization, and a linear control law is derived analytically to
achieve synchronization of the chaotic fractional-order WINDMI system. Numerical simulations
validate the main results of this work.


---

## Body

## 1. Introduction

The solar-wind-driven magnetosphere-ionosphere is a complex driven-damped dynamical system which exhibits a variety of dynamical states that include low-level steady plasma convection, episodic releases of geotail stored plasma energy into the ionosphere known broadly as substorms, and states of continuous strong unloading [1]. In 1998, Horton and Doxas [2] firstly proposed the WINDMI system, a six-dimensional nonlinear dynamics model, which was derived for the basic energy components of the night-side magnetotail coupled to the ionosphere by the region-1 currents. Smith et al. [3] explored the dynamical range of the WINDMI model. Horton et al. [1] introduced reductions to derive a new minimal three-dimensional WINDMI model. Sprott [4] further simplified the integer-order WINDMI model as follows: (1.1)dxdt=y,dydt=z,dzdt=-az-y+b-ex,
where x, y, and z are variables and a, b are positive constants. System (1.1) has a chaotic attractor when the usual parameters are a=0.7 and b=2.5, as shown in Figure 1 (see [4]).Figure 1
Phase portrait of system (1.1) versus a=0.7 and b=2.5.Fractional calculus, a generalization of differentiation and integration to an arbitrary order, is an old mathematical topic with over 300-year-old history [5]. Fractional-order differential equations can be used to describe many systems in interdisciplinary fields, but it was not used in science and engineering for many years because there exist many difficulties, such as the absence of impactful solution methods and numerical simulation schemes for fractional differential equations. However, during the past several years, with the development of computer simulation technology [6, 7], fractional calculus has been realizing the utility and applicability to various branches of science and engineering, such as fractional viscoelastic fluids [8], fractional diffusion processes [9, 10], fractional-order viscoelastic material models [11], fractional-order HIV Model [12, 13], and fractional-order controllers [14–16]. Undoubtedly, fractional calculus will be applied into more and more areas of classical and modern analysis.Chaos synchronization plays a very important role in the theory and applications. Synchronization of fractional-order chaotic systems was first presented by Deng and Li [17]. As an active research area, chaos synchronization with fractional calculus has received increasing attention in recent years due to its potentials in both theory and applications [18–25]. On the other hand, as compared with sliding mode control, standard PID feedback control, and so on, the advantage of linear state error feedback control is that it is linear and easier to implement for chaos synchronization.The remainder of this paper is organized as follows. In Section2, the WINDMI system is generalized from integer to noninteger order. Then, chaos synchronization via linear feedback control is studied in Section 3. Numerical simulations are presented in Section 4, and finally conclusions in Section 5 close the paper.


## 2. System Description

There are several definitions of fractional derivatives. In this paper, we will adopt the Riemann-Liouville fractional derivative definition as follows.Definition 2.1 (see [26]).
Theqth-order fractional derivative of function f(t) with respect to t and the terminal value 0 is given by
(2.1)dqf(t)dtq=1Γ(m-q)dmdtm∫0tf(τ)(t-τ)q-m+1dτ,
where Γ(·) is a Gamma function, m is an integer and satisfies m-1<q<m∈Z+.The integer derivative of a function has relationship with only its nearby points while the fractional derivative takes into account nonlocal characteristics like “infinite memory” [26, 27]. As a result, a model described with fractional derivative possesses memory, which may help to have a better understanding to the importance of assembling large substorm databases with a large range of event sizes. Here, we introduce fractional calculus into system (1.1). The new system is described with fractional derivative as follows: (2.2)dq1xdtq1=y,dq2ydtq2=z,dq3zdtq3=-az-y+b-ex,
in which x, y, and z are variables and a, b are positive constants; q=(q1,q2,q3) is subject to 0<q1,q2,q3<1. If q=(1,1,1), system (2.2) degenerates into system (1.1).In order to observe the synchronization behavior in two identical fractional-order WINDMI systems, we set a drive-response configuration with a drive system given by the fractional-order WINDMI systems (with three state variables denoted by the subscriptm) and with a response system (with three variables denoted by the subscript s) as follows.The drive system is described by(2.3)dq1xmdtq1=ym,dq2ymdtq2=zm,dq3zmdtq3=-azm-ym+b-exm,
and the response system is given by(2.4)dq1xsdtq1=ys+u1,dq2ysdtq2=zs+u2,dq3zsdtq3=-azs-ys+b-exs,
where u1 and u2 are the linear state error feedback controllers. Then, the rest of our task is to design suitable linear controllers that can synchronize systems (2.3) and (2.4).


## 3. Synchronization Scheme

Matignon [28] defined the internal and external stability properties of linear fractional-order differential systems of finite dimension and derived the necessary and sufficient conditions.Theorem 3.1 (see [28]).
Consider that the followingn-dimensional linear fractional-order autono-mous system:
(3.1)dqxdtq=Ax,x(0)=x0,
with 0<q<1, x∈Rn, and A∈Rn×n, is asymptotically stable if and only if |arg(λ)|>qπ/2 is satisfied for all eigenvalues λ of matrix A. Also system (3.1) is stable if and only if |arg(λ)|≥qπ/2 is satisfied for all eigenvalues λ of matrix A.Theorem 3.2 (see [29]).
Consider the following autonomousn-dimensional nonlinear fractional differential equation:
(3.2)dqX(t)dtq=F(X(t)),X(0)=X0=(x10,x20,…,xn0)T,m-1<q<m∈Z+.
Letting X̂=(x̂1,x̂2,…,x̂n)T be an equilibrium of system (3.2), that is, dqX̂/dtq=F(X)=0, and letting A=(∂F/∂X)|X=X̂ be the Jacobian matrix at the point X̂, then the point X̂ is asymptotically stable when |arg(eig(A))|>qmπ/2, where qm=max1≤i≤n{qi}.From Theorem3.2, the following corollary holds.Corollary 3.3.
LettingX̂=(x̂1,x̂2,…,x̂n)T be an equilibrium of system (3.2), that is, dqX̂/dtq=F(X̂)=0, and letting A=(∂F/∂X)|X=X̂ be the Jacobian matrix at the point X̂, then the point X̂ is locally asymptotically stable if A is an upper or lower triangular matrix and all eigenvalues of A are negative real numbers.Theorem 3.4.
The drive system (2.3) and the response system (2.4) will approach global synchronization for any initial condition with the following control law:
(3.3)u1=c1(xs-xm)-(ys-ym),u2=c2(ys-ym)-(zs-zm),
where c1<0 and c2<0.Proof.
Set the synchronization error variables as follows:(3.4)e1=xs-xm,e2=ys-ym,e3=zs-zm.
By subtracting (2.3) from (2.4) and using (3.4), the synchronization error system can be obtained as follows:
(3.5)dq1e1dtq1=e2+u1,dq2e2dtq2=e3+u2,dq3e3dtq3=-ae3-e2+exm(1-ee1),
where a>0, xm is the state variable of system (2.3), u1 and u2 are linear state error feedback controllers.
Whenu1=c1(xs-xm)-(ys-ym) and u2=c2(ys-ym)-(zs-zm), system (3.5) can be rewritten as
(3.6)dq1e1dtq1=c1e1,dq2e2dtq2=c2e2,dq3e3dtq3=-ae3-e2+exm(1-ee1).
System (3.6) has only one equilibrium point at E0=(0,0,0). Its Jacobian matrix evaluated at equilibrium point E0 is given by
(3.7)J(E0)=(c1000c20-exm+e1-1-a)=(c1000c20-exm-1-a).
Obviously, (3.7) is a lower triangular matrix, and xm is a state variable of drive system (2.3). Thus, from Corollary 3.3, one can get that xm has no effect on the stability of system (3.6), and if c1<0 and c2<0, then system (3.6) is asymptotically stable; that is, the drive system (2.3) and the response system (2.4) are synchronized finally.
The theorem is proved.Evidently, the advantage of the linear feedback controllers proposed in Theorem3.4 is that they are robust, linear, and have lower dimensions than that of the states; morever, they are easier to be designed and implemented for chaos synchronization than standard PID feedback controllers, sliding mode controllers, nonlinear feedback controllers, and so on.


## 4. Numerical Simulations

### 4.1. Discretization Scheme

Based on the Adams-Bashforth-Moulton predictor-corrector scheme [6, 30], we can build the numerical calculation formula for the proposed synchronization scheme as follows.Letting(x,y,z) and (x̂,ŷ,ẑ) represent (xm,ym,zm) and (xs,ys,zs), respectively. Setting h=T/N,tn=nh,n=0,1,…,N∈Z+, the drive system (2.3) and the response system (2.4) can be discretized as(4.1)xn+1=x0+hq1Γ(q1+2)yn+1p+hq1Γ(q1+2)∑j=0nα1,j,n+1yj,yn+1=y0+hq2Γ(q2+2)zn+1p+hq2Γ(q2+2)∑j=0nα2,j,n+1zj,zn+1=z0+hq3Γ(q3+2)(-exn+1p-yn+1p-azn+1p+b)+hq3Γ(q3+2)∑j=0nα3,j,n+1(-exj-yj-azj+b),x̂n+1=x̂0+hq1Γ(q1+2)(yn+1p+c1(x̂n+1p-xn+1p))+hq1Γ(q1+2)∑j=0nα1,j,n+1(yj+c1(x̂j-xj)),ŷn+1=ŷ0+hq2Γ(q2+2)(zn+1p+c2(ŷn+1p-yn+1p))+hq2Γ(q2+2)∑j=0nα2,j,n+1(zj+c2(ŷj-yj)),ẑn+1=ẑ0+hq3Γ(q3+2)(-ex̂n+1p-ŷn+1p-aẑn+1p+b)+hq3Γ(q3+2)∑j=0nα3,j,n+1(-ex̂j-ŷj-aẑj+b),
where (4.2)xn+1p=x0+1Γ(q1)∑j=0nβ1,j,n+1yj,yn+1p=y0+1Γ(q2)∑j=0nβ2,j,n+1zj,zn+1p=z0+1Γ(q3)∑j=0nβ3,j,n+1(-exj-yj-azj+b),x̂n+1p=x̂0+1Γ(q1)∑j=0nβ1,j,n+1(yj+c1(x̂j-xj)),ŷn+1p=ŷ0+1Γ(q2)∑j=0nβ2,j,n+1(zj+c2(ŷj-yj)),ẑn+1p=ẑ0+1Γ(q3)∑j=0nβ3,j,n+1(-ex̂j-ŷj-aẑj+b),αi,j,n+1={nqi+1-(n-qi)(n+1)qi,j=0,(n-j+2)qi+1+(n-j)qi+1-2(n-j+1)qi+1,1≤j≤n,i=1,2,3,1,j=n+1,βi,j,n+1=hqiqi((n-j+1)qi-(n-j)qi),0≤j≤n,i=1,2,3.


### 4.2. Numerical Results

Based on the above mentioned discretization scheme, the drive system (2.3) and the response system (2.4) are integrated numerically with the fractional orders q1=0.77,q2=0.98,q3=0.8 and using the initial values xm(0)=0,ym(0)=0.8,zm(0)=0,xs(0)=4,ys(0)=4.8, and zs(0)=4. Let a=0.8,b=6.4,c1=-1,c2=-1; the chaotic attractor of the drive system (2.3) is shown in Figure 2(a), and the chaotic attractor of the response system (2.4) is shown in Figure 2(b). From Figures 3(a)–3(d), it is clear that the synchronization is achieved for all these values.Chaotic attractors of drive system (2.3) and response system (2.4) versus q1=0.77,q2=0.98,q3=0.8,a=0.8,b=6.4,c1=-1, and c2=-1.
(a)
Drive system(2.3)(b)
Response system(2.4)Synchronization errors between drive system (2.3) and response system (2.4) versus q1=0.77,q2=0.98,q3=0.8,a=0.8,b=6.4,c1=-1, and c2=-1.
(a)
Synchronization errors between drive system  (2.3) and response system(2.4)(b)
Time evolutions ofxs and xm(c)
Time evolutions ofys and ym(d)
Time evolutions ofzs and zm


## 4.1. Discretization Scheme

Based on the Adams-Bashforth-Moulton predictor-corrector scheme [6, 30], we can build the numerical calculation formula for the proposed synchronization scheme as follows.Letting(x,y,z) and (x̂,ŷ,ẑ) represent (xm,ym,zm) and (xs,ys,zs), respectively. Setting h=T/N,tn=nh,n=0,1,…,N∈Z+, the drive system (2.3) and the response system (2.4) can be discretized as(4.1)xn+1=x0+hq1Γ(q1+2)yn+1p+hq1Γ(q1+2)∑j=0nα1,j,n+1yj,yn+1=y0+hq2Γ(q2+2)zn+1p+hq2Γ(q2+2)∑j=0nα2,j,n+1zj,zn+1=z0+hq3Γ(q3+2)(-exn+1p-yn+1p-azn+1p+b)+hq3Γ(q3+2)∑j=0nα3,j,n+1(-exj-yj-azj+b),x̂n+1=x̂0+hq1Γ(q1+2)(yn+1p+c1(x̂n+1p-xn+1p))+hq1Γ(q1+2)∑j=0nα1,j,n+1(yj+c1(x̂j-xj)),ŷn+1=ŷ0+hq2Γ(q2+2)(zn+1p+c2(ŷn+1p-yn+1p))+hq2Γ(q2+2)∑j=0nα2,j,n+1(zj+c2(ŷj-yj)),ẑn+1=ẑ0+hq3Γ(q3+2)(-ex̂n+1p-ŷn+1p-aẑn+1p+b)+hq3Γ(q3+2)∑j=0nα3,j,n+1(-ex̂j-ŷj-aẑj+b),
where (4.2)xn+1p=x0+1Γ(q1)∑j=0nβ1,j,n+1yj,yn+1p=y0+1Γ(q2)∑j=0nβ2,j,n+1zj,zn+1p=z0+1Γ(q3)∑j=0nβ3,j,n+1(-exj-yj-azj+b),x̂n+1p=x̂0+1Γ(q1)∑j=0nβ1,j,n+1(yj+c1(x̂j-xj)),ŷn+1p=ŷ0+1Γ(q2)∑j=0nβ2,j,n+1(zj+c2(ŷj-yj)),ẑn+1p=ẑ0+1Γ(q3)∑j=0nβ3,j,n+1(-ex̂j-ŷj-aẑj+b),αi,j,n+1={nqi+1-(n-qi)(n+1)qi,j=0,(n-j+2)qi+1+(n-j)qi+1-2(n-j+1)qi+1,1≤j≤n,i=1,2,3,1,j=n+1,βi,j,n+1=hqiqi((n-j+1)qi-(n-j)qi),0≤j≤n,i=1,2,3.


## 4.2. Numerical Results

Based on the above mentioned discretization scheme, the drive system (2.3) and the response system (2.4) are integrated numerically with the fractional orders q1=0.77,q2=0.98,q3=0.8 and using the initial values xm(0)=0,ym(0)=0.8,zm(0)=0,xs(0)=4,ys(0)=4.8, and zs(0)=4. Let a=0.8,b=6.4,c1=-1,c2=-1; the chaotic attractor of the drive system (2.3) is shown in Figure 2(a), and the chaotic attractor of the response system (2.4) is shown in Figure 2(b). From Figures 3(a)–3(d), it is clear that the synchronization is achieved for all these values.Chaotic attractors of drive system (2.3) and response system (2.4) versus q1=0.77,q2=0.98,q3=0.8,a=0.8,b=6.4,c1=-1, and c2=-1.
(a)
Drive system(2.3)(b)
Response system(2.4)Synchronization errors between drive system (2.3) and response system (2.4) versus q1=0.77,q2=0.98,q3=0.8,a=0.8,b=6.4,c1=-1, and c2=-1.
(a)
Synchronization errors between drive system  (2.3) and response system(2.4)(b)
Time evolutions ofxs and xm(c)
Time evolutions ofys and ym(d)
Time evolutions ofzs and zm


## 5. Conclusion

In this paper, we introduce fractional-order calculus into the WINDMI system. Chaos synchronization of identical master and slave fractional-order WINDMI systems is studied by utilizing linear state error feedback control technique. Based on the stability theory of nonlinear fractional-order systems, linear feedback control law for chaos synchronization has been investigated. Numerical simulations are given to verify the effectiveness of the proposed synchronization scheme.



---

*Source: 859685-2010-09-06.xml*