混沌、代换系统与Feigenbaum 现象

This item is relative to chaos、substitution systems and Feigenbaum.phenomenon in the field of dynamical systems. The several respects are all the.popular subject. The item group investigated the relations between.distributional chaos and other notions to illustrate the complex degree of a system. It has been proved that distributional chaos and Li-Yorke chaos are not generally equivalent, that an interval map with positive topological entropy has a distributionally scrambled set whose points are almost periodic, and that a minimal substitution system of constant length must be not chaotic etc. The notion of distributional chaos in a sequence was introduced. It was proved that this notion is not equivelent to distributional chaos. In adition, a proof for the existence of p order Feigenbaum maps was given. These results have important theoretical significance for revealing the internal law of chaotic motion, and for clearing the substance of chaotic motion and the mathematical basis of Feigenbaum’s phenomenon.

本项目涉及动力系统中混沌、代换系统与Feigenbaum现象等几个方面。重点考察分布混沌与其它各种刻划系统复杂程度的概念之间的关系，代换系统的动力性态以及Feigenbaum映射拟极限集的存在性等问题。本项目研究对于揭示混沌运动的内在规律，弄清混沌运动的本质以及Feigenbaum现象的数学基础具有重要的理论意义。