扩充的Ši'lnikov定理及其应用研究

Chaos is a hot research direction of nonlinearly scientific field, and chaos theory has been successfully applied to a number of fields such as engineering and secret communication. A seeming-simple yet far-unsolved problem is under what conditions chaos in the Ši'lnikov sense can be generated in 3-dimensional autonomous dynamical systems of polynomial type. This project will mainly study this issue, focusing on generation mechanisms and classification of chaos. First, we will extend the Ši'lnikov homoclinic (heteroclinic) theorem in the non-degenerate case to the degenerate and critical cases, and establish the corresponding Ši'lnikov homoclinic (heteroclinic) theorems. Second, we will study the issue of how to classify chaos in 3-dimensional autonomous dynamical systems of polynomial type, including finding those systems that have the simplest forms but can generate chaos in the sense of the Ši'lnikov homoclinic or heteroclinic theorem, and those that have the simplest forms but can generate chaos in both senses as well as those of the simplest forms where chaos is generated through other mechanisms. Through such systemic study, we try to establish a set of theories and methods of how chaos is generated in general autonomous dynamical systems of polynomial type, so as to lay a solid theoretical foundation for further applications of chaos.

混沌是非线性科学领域的一个热门研究方向，且相关理论已成功应用于工程、保密通信等众多领域。一个看似简单但远未解决的问题是：三维自治多项式型动力系统在什么条件下会产生混沌？本项目将针对这一问题开展研究，聚焦于Ši'lnikov意义下混沌的产生机制与分门别类。首先研究如何推广非退化情形时的Ši'lnikov同（异）宿轨定理到临界或退化情形，并建立相应情形时的Ši'lnikov同（异）宿轨定理；其次研究三维自治多项式型动力系统中混沌的分类问题，包括给出产生Ši'lnikov同宿或异宿意义下混沌的最简系统形式，产生这两种意义下的所谓混合型混沌的最简系统形式，以及通过其它机制产生混沌的最简系统形式。通过系统研究，本项目试图建立一套研究一般自治多项式型动力系统中混沌的理论与方法，为混沌的进一步应用奠定理论基础。

本项目研究三维自治多项式型动力系统的混沌生成机制问题。主要包括：研究如何推广非退化情形时的Ši'lnikov同（异）宿轨定理到临界或退化情形，其中我们依据Ši'lnikov定理，利用相平面分析、动力学分叉分析等理论和数值手段具体分析了一个典型生物系统的动力学行为；其次研究三维自治多项式型动力系统中混沌的分类问题，借用生物学上的调控原理将三维二次自治型混沌系统映射为调控网络，讨论网络结构分类与混沌分类之间的关系，从调控网络结构的观点提出关于三维二次自治型混沌系统的分类准则。