We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method. 1. Introduction Chaos synchronization is the concept of closeness of the frequencies between different periodic oscillations generated by two chaotic systems, one of which is the master and the other is the slave. Since the pioneering work of Pecora and Carroll [1] who proposed a method to synchronize two identical chaotic systems, chaos synchronization has attracted a lot of attention in a variety of research fields over the last two decades. This is because chaos synchronization can be used in many areas such as physics, engineering, and particularly in secure communication [2–5]. Many methods have been proposed to synchronize chaotic systems including active control [6], back-stepping control [7], linear feedback control [8], adaptive control theory [9], sliding mode control [10, 11], and fuzzy control [12]. For example, Bhalekar and Daftardar-Gejji [13] used active control for the problem of synchronization of fractional-order Liu system with fractional-order Lorenz system. Based on the idea of tracking control and stability theory of fractional-order systems, Zhou and Ding [14] designed a controller to synchronize the fractional-order Lorenz chaotic system via fractional-order derivative. Zhang and Yang [15] dealt with the lag synchronization of fractional-order chaotic systems with uncertain parameters. Projective synchronization of a class of fractional-order hyperchaotic system with uncertain parameters was studied by Bai et al. [16] as well, but the derivative orders of the state in response system was the same with drive system. Chen et al. [17] designed a sliding mode controller for a class of fractional-order chaotic systems. However, most of the above-mentioned work on chaos synchronization has focused on fractional-order chaos and integer-order systems, respectively. To the best of our knowledge, there has been little information available about the synchronization between
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