Chaotic synchronization between different fractional-order chaotic systems

https://doi.org/10.1016/j.jfranklin.2011.09.004Get rights and content

Abstract

Based on the idea of tracking control and stability theory of fractional-order systems, a novel synchronization approach for fractional order chaotic systems is proposed. We prove that the synchronization between drive system and response system with different fractional order q can be achieved, and the synchronization between different fractional-order chaotic systems with different fractional order q can be achieved. Two examples are used to illustrate the effectiveness of the proposed synchronization method. Numerical simulations coincide with the theoretical analysis.

Introduction

In 1990, Pecora and Carroll presented the concept of “chaotic synchronization” for the first time [1] and introduced a method to synchronize two identical chaotic systems with different initial conditions [2]. Chaos synchronization has become an active research subject in nonlinear science because of its potential applications in secure communication [3], [4], neuron systems [5], [6], [7], the study of laser dynamics [8] and control processing [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].

In recent years, study on the dynamics of fractional-order differential systems has attracted interest of many researchers. It is demonstrated that some fractional-order differential systems behave chaotically or hyperchaotically, such as the fractional-order Lorenz system, the fractional-order Chua's system, the fractional Rossler system, the fractional modified Duffing system, the fractional order unified system and the Chen system [21], [22], [23], [24], [25]. On the other hand, chaos synchronization for the fractional order systems are just beginning to attract some attention due to its potential applications in secure communication and control processing [25], [26], [27], [28], [29], [30]. More recently, many authors begin to investigate the dynamics and synchronization of fractional-order chaotic systems [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34].

However, almost all existed synchronization methods for fractional-order chaotic systems mainly focus on that, which has the same fractional order q for drive fractional-order chaotic system and response fractional-order chaotic system. So far, there has been little information available in literature about the drive fractional-order chaotic system and response fractional-order chaotic system with different fractional order q, and little information available in literature about two different fractional-order chaotic systems with different fractional order q.

Motivated by all above works, in this paper, one synchronization method for fractional-order chaotic systems, which is different from the previous works, is given for the drive fractional-order chaotic system and response fractional-order chaotic system with different fractional order q, and for two different fractional-order chaotic systems with different fractional order q. The proposed technique is based on the idea of tracking control and stability theory of fractional order systems, which is simple and theoretically rigorous. Two groups of examples are considered and their numerical simulations are performed. The first example is the synchronization of fractional-order Chen chaotic system, in which the drive fractional-order chaotic system and response fractional-order chaotic system are the different fractional order q. The second example is the synchronization between fractional-order Lorenz chaotic system and fractional-order Chen chaotic system with different fractional order q. Numerical simulations coincide with the theoretical analysis.

Section snippets

Problem formulation

There are many definitions of fractional derivatives. In the following, we introduce the most common one of themdqf(t)/dtqDqf(t)=Jmqf(m)(t),q>0,where m is the first integer, which is not less than q, Dq is generally called “q-order Caputo differential operator” [30], f(m)(t) is the m-order derivative in the usual sense, and Jq(q>0) is the q-order Riemann–Liouville integral operator with expressionJqf(t)=1Γ(q)0t(tτ)q1f(τ)dτ,where Γ(.) is the gamma function.

Now, we consider the following n

Application of the above-mentioned scheme

To illustrate the effectiveness of the proposed synchronization scheme, two groups of examples are considered and their numerical simulations are performed.

First, consider the synchronization of fractional-order Chen chaotic system [36], and the drive fractional-order chaotic system and response fractional-order chaotic system with different fractional order.

The Chen system, introduced by Chen and Ueta in 1999 [36], is similar but not topologically equivalent to the Lorenz system and is

Conclusions

In this paper, we investigated the synchronization of fractional order chaotic systems based on the idea of tracking control and stability theory of fractional-order systems. The derived method in the present paper shows that the synchronization between drive system and response system with different fractional order q can be achieved, and the synchronization between different fractional-order chaotic systems with different fractional order q can be achieved. Numerical experiments show that the

Acknowledgements

This work is supported by Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ110525.

References (38)

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