分段光滑系统含周期轨的奇异环的存在性及其诱导的奇异马蹄与分岔

Singular cycle is one of the mechanisms leading to complicated dynamic behaviors of dynamical systems, thus the existence of singular cycles and the phenomena of chaos and bifurcation induced by them is one of the core topics in the study of dynamical systems, however, most of the current studies involve the situation that singular elements contain only equilibrium points. This project investigates the existence of singular cycles whose singular elements include periodic orbits and the relevant phenomena of chaos and bifurcation for high dimensional piecewise smooth dynamical systems. By using some fundamental theories of dynamical systems, such as the linearization of vector fields, the discretization of flows, the singular horseshoe theory, the homoclinic (heteroclinic) bifurcation and so on, the project studies the following contents: the geometric relations of the invariant manifolds of subsystems and the existence and construction of singular cycles containing periodic orbits; the study of the chaotic phenomena induced by the singular cycles, especially the mechanisms leading to singular horseshoe chaos; the bifurcation study of the singular cycles, including the emerge and disappearance of chaos, and the existence, stability and number of periodic orbits generated by singular cycles bifurcation. The study of this project can further contribute to the qualitative theory of piecewise smooth systems, and also provide some theoretical basis for the construction of piecewise smooth systems satisfying the specific conditions in practical application.

奇异环是导致动态系统复杂动力学行为的重要机制之一，因此奇异环的存在性及其诱导的混沌与分岔现象的理论研究是动力系统研究的核心课题之一，但当前研究涉及较多的是奇异元仅含平衡点的情况。本项目针对高维分段光滑系统，研究奇异元含周期轨的奇异环的存在性及其诱导的混沌与分岔现象。通过综合利用向量场的线性化、流的离散化、奇异马蹄理论、同(异)宿分岔等动态系统基础理论，研究如下具体内容：子系统不变流行的几何关系与奇异元含周期轨的奇异环的存在性及其构造；奇异元含周期轨的奇异环诱导的混沌现象研究，特别是奇异马蹄混沌的产生机制；奇异元含周期轨的奇异环分岔研究，如混沌不变集的产生与消失，奇异环分岔导致的周期轨存在性、稳定性及个数等。通过本项目的研究，可以进一步完善分段光滑系统定性理论，也可为实际应用中构造满足特定条件的分段光滑系统提供一定的理论依据。

奇异环是导致非光滑系统动态行为复杂性的关键机制之一，奇异环的存在性以及由其诱导的分岔行为研究是目前非光滑系统研究的热点问题。本项目针对分段光滑系统，集中研究与奇异环相关的如下重要问题：奇异元含周期轨的奇异环的存在性，含有单个或者多个切换流形的系统的奇异环的存在性及其诱导的奇异环分岔现象，高维系统双焦点异宿环诱导的拓扑马蹄混沌等一些列重要的复杂动力学现象。得到了一类三维分段光滑系统连接鞍形周期轨道和鞍平衡点的异宿环存在的充分条件，一类三维分段仿射系统连接鞍平衡点的奇异环的存在性及其分岔诱导的周期解的存在性及稳定性，一类含有双切换平面的三维分段仿射系统异宿环的存在性及其诱导的混沌不变集的数值实现，一类四维分段仿射系统由双焦点异宿环诱导的可数无穷多拓扑马蹄的存在性等一系列重要结果。本项目的研究成果揭示了分段光滑系统发生周期行为、混沌行为的一些内在机制，丰富了非光滑系统的理论体系，为进一步研究分段光滑系统在工程技术和自然科学领域的应用奠定了一定的理论基础。