一类Caputo型分数阶混沌系统及其同步研究

The fractional chaotic system is a new chaotic system, and it is better than integer order chaotic system in complexity because of the particularity about the definitions of fractional calculus and multiple parameters in system. For example, the fractional chaotic system can better mask information to increase security, while the sender and receiver should be synchronous during the implementation of communication based on chaos. So, it is very useful for studying fractional chaotic system and its synchronization. Based on the development of fractional chaotic system and its synchronization，together with the stability theory of fractional differential system, this project will further study a class of Caputo type fractional chaotic systems with orders lying in (1,2) and their synchronization. Furthermore, the order scales which the corresponding fractional systems can lead to chaotic behavior will be also derived.

分数阶混沌系统是一类新型的混沌系统，由于分数阶微积分定义的特殊性以及系统中的多参数，使得它在复杂度上更优于经典的整数阶混沌系统，如分数阶混沌系统能够更好地掩盖信息来增强通信中的安全性；而实时混沌通信的实现必须要求接收和发送双方保持同步，因此，研究分数阶混沌系统及其同步有着非常重要的意义。本项目基于分数阶混沌系统的研究现状，拟结合分数阶微分系统的稳定性理论，进一步研究一类阶介于1、2之间的Caputo型分数阶混沌系统及其同步问题，并推导分数阶系统产生混沌的阶的范围。

本项目基于解的存在性和稳定性理论, 研究了一类Caputo型分数阶微分系统的复杂动力学行为，包括混沌和同步现象。首先，讨论了一类阶大于1的微分方程的解在复数域内的存在性。利用数值方法，结合已有的稳定性理论和分数阶微积分的特殊性质，发现阶位于1与2之间的Caputo型分数阶Chua系统存在混沌现象。然后针对不同的耦合模式，包括one-way 模式和Pecora-Carroll模式，将稳定性理论应用于耦合系统对应的误差系统，进一步研究此类阶介于1、2之间的Caputo型分数阶混沌系统的同步。并采用数值模拟的方法分析了混沌系统的耦合参数的变化对同步快慢产生的影响。