Research On Fractional Order Chaotic Systems And Their Synchronization

The studies on fractional order chaotic systems have attracted increasing atten-tions in recent years. And it is well-known that the predictor-corrector scheme is only valid for the numerical solutions of the fractional order chaotic systems. Besides, chaos synchronization of the fractional order systems becomes a challenging and interesting problem due to its potential applications in secure communication and control process-ing. However, the chaotic synchronization methods are still limited. And few results about the synchronization of different dimensional fractional order chaotic systems are shown. In this paper, we mainly focus on the schemes for the solutions and synchro-nization of the fractional order chaotic systems. The main achievements are as follows:1. The application of the homotopy perturbation method (HPM) for the analytical solutions of the fractional order chaotic system is a new topic. It is treated as an algo-rithm in a sequence of intervals for finding accurate approximate solutions to the dy-namic systems. We call this technique as the multistage homotopy perturbation method (MHPM). In particular, we look at the accuracy of the new improved MHPM for the fractional order chaotic Chen system and hyperchaotic Lorenz system. Numerical re-sults show that the new technique is a simple, yet powerful analytic-numeric scheme for handling the problems of the fractional order chaotic systems.2. A new method is provided for the hybrid synchronization of the different di-mensional fractional order chaotic systems. It is shown that the slave system can be synchronized with the projection of the master system generated through states trans-formation. Based on the stability theorem of linear fractional order systems, a suit-able controller for achieving the synchronization is given. The hybrid synchronization between the fractional order chaotic system and hyperchaotic system is successfully achieved in both reduced order and increased order. The corresponding numerical re-sults verify the effectiveness of the proposed method.