分段光滑系统的不变流形结构与动力学分析

项目名称： 分段光滑系统的不变流形结构与动力学分析

项目编号： No.11472111

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 杨晓松

作者单位： 华中科技大学

项目金额： 75万元

中文摘要： 分段光滑动力系统理论有很强的应用背景和重要的理论意义，是动力学与控制领域的新兴分支。本项目研究下面几个重要问题：1不连续性以及非光滑性边界附近的光滑子系统的不变流形的特性及其在不连续边界的几何关系导致周期解或极限环存在的问题,以及在两个中心流形不匹配的情况下新的周期解分叉现象。2 三维和四维分段光滑向量场不变流形的特性及其在不连续性边界的几何关系导致混沌存在的机理，尤其是周期轨分叉和同宿轨的存在性，并针对分段仿射系统研究不变流形的结构与周期解及同宿分叉问题。3 发展复合映射的不变集和不变流形理论，并借助于非光滑或不连续性边界诱导的子庞卡莱映射的复合来探讨分段光滑向量场的不变流形，研究同宿点的存在性和混沌产生机制。通过系统地研究上述问题，揭示分段光滑系统的不变子流形的结构和局部子流形在不连续性边界附近的相互关系及其导致周期和混沌动力学行为的机理，以期为深入研究分段光滑系统打下一定的理论基础。

中文关键词： 非线性动力学；分段光滑系统；不变流形；周期分叉；同宿点与混沌

英文摘要： Piecewise-smooth dynamical systems are of theoretical significance and have broad applications in practical problems. The study of piecewise-smooth dynamical systems is a relatively new area in dynamics and control. This project studies the following important problems:1 The geometric properties of invariant manifolds of sub-smooth systems in the vicinity of discontinuity and nonsmoothness boundary and their role in existence of periodic solutions and limit cycles, and new periodic bifurcations. 2 The geometric properties of invariant manifolds of sub-smooth systems and the existence of chaos as well as periodic bifurcations and homoclinic orbits. 3 Establish a theory for invariant sets and invariant submanifolds of composite maps, and study invariant manifolds of piecewise smooth vector fields by means of the composite maps induced by sub-Poincaré maps defined on discontinuity and nonsmoothness boundary and the mechanism of existence of homoclinic points and chaos. The studies of the above problems can reveal the relations between the geometric properties of invariant manifolds of sub-smooth systems in the vicinity of discontinuity and nonsmoothness boundary and existence of chaos as well as periodic solutions and limit cycles, and present a theory for further studies on piecewise-smooth dynamical systems.

英文关键词： nonlinear dynamics;piecewise-smooth system;invariant manifold;periodic bifurcation;homoclinic point and chaos