一簇Lorenz映射的混沌行为的符号动力学分析

Analyzing the Chaotic Behavior of a Family of Lorenz Maps by Symbolic Dynamics

首先从符号动力学的角度论证了一簇Lorenz映射且有的混沌性质:稠密的周期轨道,周期的集合,拓扑熵,几乎所有(关于Lebesgue测度)的点的Lyapunov指数;并从揉序列的分析给出了该簇映射的拓扑熵的一个下界及Lyapunov指数的一个下界与上界,在很大程度上反应了Lorenz系统的复杂程度.其次仍从符号动力学的角度论证了更一般的Lorenz映射,通过设立参数空间,穷尽了Lorenz映射中函数为直线段的所有情况,并得出同前述Lorenz映射相似的且较为复杂的性质.

At first, the paper gives proof to some chaotic properties which characterize a family of Lorenzmaps. The properties include dense periodic obits, set of periodicals, positive topological entropy and positive local Lyapunov exponent for almost points. The theorems give a lower bound for the topological entropy and both the lower and upper bounds for the local Lyapunov exponent which virtually characterizes the complexity of a nonlinear dynamic system. The process of proof quite simplifies the one by topology. Then in a broad sense the paper studies a more general family of Lorenz maps consisting only of lines by ranging the parameter in the parameter space and finally similar properties to the above Lorenz maps are also resulted.