一种异结构分数阶混沌系统投影同步的新方法

A new method for projective synchronization of different fractional order chaotic systems

基于Lyapunov稳定性理论和分数阶系统稳定理论以及分数阶非线性系统性质,提出了一种用来判定分数阶混沌系统是否稳定的新的判定定理,并把该理论运用于对分数阶混沌系统的控制与同步,同时给出了数学证明过程,严格保证了该方法的正确性与一般适用性.运用所提出的稳定性定理,实现了异结构分数阶混沌系统的投影同步.对分数阶Lorenz混沌系统与分数阶Liu混沌系统实现了投影同步;针对四维超混沌分数阶系统,也实现了异结构投影同步.该稳定性定理避免了求解分数阶平衡点以及Lyapunov指数的问题,从而可以方便地选择出控制律,并且所得的控制器结构简单、适用范围广.数值仿真的结果取得了预期的效果,进一步验证了这一稳定性定理的正确性及普遍适用性.

Based on the Lyapunov theory as the breakthrough point, and based on the fractional order system stability theory and properties of fractional nonlinear system, a kind of fractional-order chaotic system is proposed to determine whether the new theorem is stable, and the theory is used for fractional order control and synchronization of chaotic systems, and gives a mathematical proof process to strictly ensure the correctness of the method and general applicability. Then the proposed stability theorem is used to achieve the projective synchronization of fractional Lorenz chaotic system with fractional order chaotic Liu system, as well as the projective synchronization of four-dimensional hyperchaos of fractional order systems of different structures. In the stability theorem solving the fractional balance point and the Lyapunov index are avoided, therefore a control law can be easily selected, and the obtained controller has the advantages of simple structure and wide range of application. Finally, the expected numerical simulation results are achieved, which further proves the correctness and universal applicability of the stability theorem.