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

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.

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: 𝑑𝑥𝑑𝑡=𝑦,𝑑𝑦𝑑𝑡=𝑧,𝑑𝑧𝑑𝑡=−𝑎𝑧−𝑦+𝑏−𝑒^𝑥, (1.1) 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 Section 2, 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]). The 𝑞th-order fractional derivative of function 𝑓(𝑡) with respect to 𝑡 and the terminal value 0 is given by 𝑑^𝑞𝑓(𝑡)𝑑𝑡^𝑞=1𝑑Γ(𝑚−𝑞)𝑑𝑡^𝑚𝑚𝑡₀𝑓(𝜏)(𝑡−𝜏)^{𝑞−𝑚+1}𝑑𝜏, (2.1) 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: 𝑑^𝑞1𝑥𝑑𝑡^𝑞1𝑑=𝑦,_𝑞2𝑦𝑑𝑡^𝑞2𝑑=𝑧,_𝑞3𝑧𝑑𝑡^𝑞3=−𝑎𝑧−𝑦+𝑏−𝑒^𝑥, (2.2) 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 subscript 𝑚) and with a response system (with three variables denoted by the subscript 𝑠) as follows.

The drive system is described by 𝑑^𝑞1𝑥_𝑚𝑑𝑡^𝑞1=𝑦_𝑚,𝑑^𝑞2𝑦_𝑚𝑑𝑡^𝑞2=𝑧_𝑚,𝑑^𝑞3𝑧_𝑚𝑑𝑡^𝑞3=−𝑎𝑧_𝑚−𝑦_𝑚+𝑏−𝑒^𝑥𝑚, (2.3) and the response system is given by𝑑^𝑞1𝑥_𝑠𝑑𝑡^𝑞1=𝑦_𝑠+𝑢₁𝑑_𝑞2,𝑦_𝑠𝑑𝑡^𝑞2=𝑧_𝑠+𝑢₂𝑑_𝑞3,𝑧_𝑠𝑑𝑡^𝑞3=−𝑎𝑧_𝑠−𝑦_𝑠+𝑏−𝑒^𝑥𝑠, (2.4)(1)(2) 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 following 𝑛-dimensional linear fractional-order autono-mous system: 𝑑^𝑞𝑥𝑑𝑡^𝑞=𝐴𝑥,𝑥(0)=𝑥₀, (3.1) with 0<𝑞<1, 𝑥∈𝑅^𝑛, and 𝐴∈𝑅^𝑛×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 autonomous 𝑛-dimensional nonlinear fractional differential equation: 𝑑^𝑞𝑋(𝑡)𝑑𝑡^𝑞=𝐹(𝑋(𝑡)),𝑋(0)=𝑋₀=(𝑥₁₀,𝑥₂₀,…,𝑥_𝑛0)^𝑇,𝑚−1<𝑞<𝑚∈𝑍⁺. (3.2) Letting 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 be an equilibrium of system (3.2), that is, 𝑑^𝑞𝑋/𝑑𝑡^𝑞=𝐹(𝑋)=0, and letting 𝐴=(𝜕𝐹/𝜕𝑋)|_𝑋=𝑋 be the Jacobian matrix at the point 𝑋, then the point 𝑋 is asymptotically stable when |arg(eig(𝐴))|>𝑞_𝑚𝜋/2, where 𝑞_𝑚=max_{1≤𝑖≤𝑛}{𝑞_𝑖}.

From Theorem 3.2, the following corollary holds.

Corollary 3.3. Letting 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 be an equilibrium of system (3.2), that is, 𝑑^𝑞𝑋/𝑑𝑡^𝑞=𝐹(𝑋)=0, and letting 𝐴=(𝜕F/𝜕X)|_𝑋=𝑋 be the Jacobian matrix at the point 𝑋, then the point 𝑋 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: 𝑢₁=𝑐₁(𝑥_𝑠−𝑥_𝑚)−(𝑦_𝑠−𝑦_𝑚),𝑢₂=𝑐₂(𝑦_𝑠−𝑦_𝑚)−(𝑧_𝑠−𝑧_𝑚), (3.3) where 𝑐₁<0 and 𝑐₂<0.

Proof. Set the synchronization error variables as follows: 𝑒₁=𝑥_𝑠−𝑥_𝑚,𝑒₂=𝑦_𝑠−𝑦_𝑚,𝑒₃=𝑧_𝑠−𝑧_𝑚. (3.4) By subtracting (2.3) from (2.4) and using (3.4), the synchronization error system can be obtained as follows: 𝑑^𝑞1𝑒₁𝑑𝑡^𝑞1=𝑒₂+𝑢₁,𝑑^𝑞2𝑒₂𝑑𝑡^𝑞2=𝑒₃+𝑢₂,𝑑^𝑞3𝑒₃𝑑𝑡^𝑞3=−𝑎𝑒₃−𝑒₂+𝑒^𝑥𝑚(1−𝑒^𝑒1), (3.5) where 𝑎>0, 𝑥_𝑚 is the state variable of system (2.3), 𝑢₁ and 𝑢₂ are linear state error feedback controllers.
When 𝑢₁=𝑐₁(𝑥_𝑠−𝑥_𝑚)−(𝑦_𝑠−𝑦_𝑚) and 𝑢₂=𝑐₂(𝑦_𝑠−𝑦_𝑚)−(𝑧_𝑠−𝑧_𝑚), system (3.5) can be rewritten as 𝑑^𝑞1𝑒₁𝑑𝑡^𝑞1=𝑐₁𝑒₁,𝑑^𝑞2𝑒₂𝑑𝑡^𝑞2=𝑐₂𝑒₂,𝑑^𝑞3𝑒₃𝑑𝑡^𝑞3=−𝑎𝑒₃−𝑒₂+𝑒^𝑥𝑚(1−𝑒^𝑒1). (3.6)
System (3.6) has only one equilibrium point at 𝐸₀=(0,0,0). Its Jacobian matrix evaluated at equilibrium point 𝐸₀ is given by 𝐽(𝐸₀𝑐)=¹000𝑐₂0−𝑒^{𝑥𝑚+𝑒1}=𝑐−1−𝑎¹000𝑐₂0−𝑒^𝑥𝑚−1−𝑎. (3.7)
Obviously, (3.7) is a lower triangular matrix, and 𝑥_𝑚 is a state variable of drive system (2.3). Thus, from Corollary 3.3, one can get that 𝑥_𝑚 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 Theorem 3.4 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 (𝑥,𝑦,𝑧) represent (𝑥_𝑚,𝑦_𝑚,𝑧_𝑚) and (𝑥_𝑠,𝑦_𝑠,𝑧_𝑠), respectively. Setting ℎ=𝑇/𝑁,𝑡_𝑛=𝑛ℎ,𝑛=0,1,…,𝑁∈𝑍⁺, the drive system (2.3) and the response system (2.4) can be discretized as𝑥_𝑛+1=𝑥₀+ℎ^𝑞1Γ(𝑞₁𝑦+2)^𝑝𝑛+1+ℎ^𝑞1Γ(𝑞₁+2)^𝑛_𝑗=0𝛼_{1,𝑗,𝑛+1}𝑦_𝑗,𝑦_𝑛+1=𝑦₀+ℎ^𝑞2Γ(𝑞₂𝑧+2)^𝑝𝑛+1+ℎ^𝑞2Γ(𝑞₂+2)^𝑛_𝑗=0𝛼_{2,𝑗,𝑛+1}𝑧_𝑗,𝑧_𝑛+1=𝑧₀+ℎ^𝑞3Γ(𝑞₃+2)(−𝑒^{𝑥𝑝𝑛+1}−𝑦^𝑝_𝑛+1−𝑎𝑧^𝑝_𝑛+1ℎ+𝑏)+^𝑞3Γ(𝑞₃+2)^𝑛_𝑗=0𝛼_{3,𝑗,𝑛+1}(−𝑒^𝑥𝑗−𝑦_𝑗−𝑎𝑧_𝑗𝑥+𝑏),_𝑛+1=𝑥₀+ℎ^𝑞1Γ(𝑞₁+2)(𝑦^𝑝_𝑛+1+𝑐₁(𝑥^𝑝_𝑛+1−𝑥^𝑝_𝑛+1ℎ))+^𝑞1Γ(𝑞₁+2)^𝑛_𝑗=0𝛼_{1,𝑗,𝑛+1}(𝑦_𝑗+𝑐₁(𝑥_𝑗−𝑥_𝑗𝑦)),_𝑛+1=𝑦₀+ℎ^𝑞2Γ(𝑞₂+2)(𝑧^𝑝_𝑛+1+𝑐₂(𝑦^𝑝_𝑛+1−𝑦^𝑝_𝑛+1ℎ))+^𝑞2Γ(𝑞₂+2)^𝑛_𝑗=0𝛼_{2,𝑗,𝑛+1}(𝑧_𝑗+𝑐₂(𝑦_𝑗−𝑦_𝑗𝑧)),_𝑛+1=𝑧₀+ℎ^𝑞3Γ(𝑞₃𝑥+2)(−𝑒^𝑝𝑛+1−𝑦^𝑝_𝑛+1−𝑎𝑧^𝑝_𝑛+1ℎ+𝑏)+^𝑞3Γ(𝑞₃+2)^𝑛_𝑗=0𝛼_{3,𝑗,𝑛+1}(−𝑒𝑥_𝑗−𝑦_𝑗−𝑎𝑧_𝑗+𝑏), (4.1) where 𝑥^𝑝_𝑛+1=𝑥₀+1Γ(𝑞₁)^𝑛_𝑗=0𝛽_{1,𝑗,𝑛+1}𝑦_𝑗𝑦_𝑝𝑛+1,=𝑦₀+1Γ(𝑞₂)^𝑛_𝑗=0𝛽_{2,𝑗,𝑛+1}𝑧_𝑗𝑧_𝑝𝑛+1,=𝑧₀+1Γ(𝑞₃)^𝑛_𝑗=0𝛽_{3,𝑗,𝑛+1}(−𝑒^𝑥𝑗−𝑦_𝑗−𝑎𝑧_𝑗𝑥+𝑏),_𝑝𝑛+1=𝑥₀+1Γ(𝑞₁)^𝑛_𝑗=0𝛽_{1,𝑗,𝑛+1}(𝑦_𝑗+𝑐₁(𝑥_𝑗−𝑥_𝑗𝑦)),_𝑝𝑛+1=𝑦₀+1Γ(𝑞₂)^𝑛_𝑗=0𝛽_{2,𝑗,𝑛+1}(𝑧_𝑗+𝑐₂(𝑦_𝑗−𝑦_𝑗𝑧)),_𝑝𝑛+1=𝑧₀+1Γ(𝑞₃)^𝑛_𝑗=0𝛽_{3,𝑗,𝑛+1}(−𝑒𝑥_𝑗−𝑦_𝑗−𝑎𝑧_𝑗𝛼+𝑏),_{𝑖,𝑗,𝑛+1}=𝑛^𝑞𝑖+1−(𝑛−𝑞_𝑖)(𝑛+1)^𝑞𝑖,𝑗=0,(𝑛−𝑗+2)^𝑞𝑖+1+(𝑛−𝑗)^𝑞𝑖+1−2(𝑛−𝑗+1)^𝑞𝑖+1𝛽,1≤𝑗≤𝑛,𝑖=1,2,3,1,𝑗=𝑛+1,_{𝑖,𝑗,𝑛+1}=ℎ^𝑞𝑖𝑞_𝑖((𝑛−𝑗+1)^𝑞𝑖−(𝑛−𝑗)^𝑞𝑖),0≤𝑗≤𝑛,𝑖=1,2,3. (4.2)(3)(4)(5)(6)(7)(8)(9)

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 𝑥_𝑚(0)=0,𝑦_𝑚(0)=0.8,𝑧_𝑚(0)=0,𝑥_𝑠(0)=4,𝑦_𝑠(0)=4.8, and 𝑧_𝑠(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.

(a) Drive system(2.3)

(b) Response system(2.4)

Figure 2 

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) Synchronization errors between drive system (2.3) and response system(2.4)

(b) Time evolutions of 𝑥𝑠 and 𝑥𝑚

(c) Time evolutions of 𝑦𝑠 and 𝑦𝑚

(d) Time evolutions of 𝑧𝑠 and 𝑧𝑚

Figure 3 

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.

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.

Acknowledgments

The authors are very grateful for the editor in charge Yuri Vladimirovich Mikhlin for the efficient editorial work. The insightful suggestions from the referees improved the quality and the readability substantially. This paper is supported partly by the China Postdoctoral Science Foundation (Grant no. 20100470783), Specialized Research Fund for Doctoral Program of Higher Education from Ministry of Education of China (Grant no. 20090032110031), and National Natural Science Foundation of China (Grant no. 60904063).