分段仿射系统奇异环的存在性与混沌

发布时间：2018-02-25 07:00

本文关键词： 混沌奇异环分段仿射系统拓扑马蹄庞加莱映射　出处：《华中科技大学》2016年博士论文　论文类型：学位论文

【摘要】：混沌现象是自然科学中广泛存在但却又十分有趣的动力学现象,在光滑动力系统中著名的Shilnikov类型的定理针对混沌不变集的存在性给出了严格的理论,这些定理部分地被推广到分段光滑系统中。但是Shilnikov类型的定理中有一个非常重要的假设条件,即同宿轨或异宿环的存在性。对于一般的系统来说,探索系统同宿轨或异宿环的存在性是非常棘手的。幸运的是,针对分段仿射系统来说,我们不仅能够显式的表示各个子系统的稳定流形和不稳定流形,还可以显式的表示各个子系统的解,因此分段仿射系统对于研究同宿轨或异宿环的存在性提供了良好的模型。在此基础上还可以讨论混沌不变集的存在性。本文正是致力于分段仿射系统同宿轨或异宿环的存在性以及混沌的研究,取得了如下创新成果：(1)三维分段仿射系统同宿轨存在性。研究了一类三维分段仿射系统的同宿轨的存在性,给出了与切换面横截相交于两点的同宿轨存在的充要条件,给出了构造混沌系统的严格的数学方法。(2)三维分段仿射系统异宿环存在性及混沌。研究了一类三维分段仿射系统异宿环的存在性,给出了与切换面横截相交于两点的异宿环存在的充要条件,并在此基础上运用拓扑马蹄理论给出了混沌不变集存在的严格证明。给出了构造混沌系统的严格的数学方法。(3)四维分段仿射系统双焦点同宿轨的存在性。研究了具有两个子系统的四维分段仿射系统双焦点同宿轨的存在性,给出了与切换面横截相交于两点的双焦点同宿轨存在的充要条件,并给出了构造混沌系统的严格数学方法。(4)四维分段仿射系统双焦点异宿环的存在性。研究了具有两个子系统的四维分段仿射系统双焦点异宿环的存在性,给出了与切换面横截相较于两点的双焦点异宿环存在的充要条件。并在此基础上,构造了一个具有双焦点异宿环的四维系统,给出了混沌不变集存在的计算机仿真结果。本文的具体内容安排如下：第一章主要介绍了分段光滑系统的一些基本概念和分段光滑动力系统的研究现状。第二章主要介绍了符号动力系统与拓扑马蹄理论。第三章主要研究了一类三维分段仿射系统同宿轨存在的充要条件,给出了一种构造混沌系统的数学方法,并在此基础上,构造了几个混沌系统,给出了相关的计算机仿真结果。第四章介绍了一类三维分段仿射系统异宿环存在的充要条件,并在此基础上运用拓扑马蹄理论证明了混沌不变集的存在性,给出了一种构造混沌系统的数学方法。并在此基础上,构造了几个混沌系统,给出了相关的计算机仿真结果。第五章是四维分段仿射系统奇异坏的存在性。首先研究了一类四维分段仿射系统双焦点同宿轨存在的充要条件,给出了一种构造混沌系统的数学方法,并在此基础上,构造了一个混沌系统,给出了相关的计算机仿真结果。其次研究了一类四维分段仿射系统双焦点异宿环存在的充要条件,构造了一个具有双焦点异宿环的四维系统,给出了混沌不变集存在的计算机仿真结果。第六章对全文的工作进行了总结,并对下一步工作拟定计划。
[Abstract]:Chaos is a dynamic phenomenon but very interesting widely exists in the natural sciences in smooth dynamical systems in the famous Shilnikov type theorem for the existence of chaotic invariant set gives the strict theory, these theorems is partially extended to piecewise smooth system. But the Shilnikov type theorem is a very important the assumption that the existence of homoclinic or heteroclinic loop. For the general systems, explore the existing system of homoclinic or heteroclinic loop is very difficult. Fortunately, the piecewise affine systems, we can not only show that the stable manifold of subsystems and unstable manifold type that can also shows the various subsystems of the solution, so the piecewise affine system for the study of homoclinic or heteroclinic existence provides a good model. On this basis can be discussed. The existence of chaotic invariant set. This paper is devoted to study the existence of chaos and the piecewise affine system homoclinic or heteroclinic loop, the main contributions are as follows: (1) three dimensional piecewise affine system existence of homoclinic orbits. Existence of a class of three-dimensional piecewise affine systems of homoclinic orbits, are given with the switching surface transverse intersection in two necessary and sufficient conditions for existence of homoclinic orbits, a strict mathematical method for constructing chaotic system is presented in this paper. (2) three dimensional piecewise affine systems and existence of heteroclinic chaos. To study a class of three-dimensional piecewise affine systems heteroclinic the existence of heteroclinic loop gives necessary and sufficient conditions with the switching surface transverse intersection at two points exist, and based on the use of topological horseshoe theory gives the chaotic invariant sets are proved strictly. The strict mathematical method of constructing chaotic system is presented in this paper. (3) the four-dimensional piecewise affine systems The existence of double focus homoclinic orbit. The existence of four with two subsystems of piecewise affine system with dual focus homoclinic orbits, and gives the necessary and sufficient conditions of the switching surface transverse intersection in double homoclinic orbits have the focus, and gives the strict mathematical method of constructing chaotic system. (4) the existence of four piecewise affine system bifocus heteroclinic loop. The existence of four with two subsystems of piecewise affine system with dual focus heteroclinic loop, gives a sufficient and necessary condition of the dual focus heteroclinic and switching surface cross section compared to the two loops exist. And on this basis, we construct a with two different focus homoclinic loop four-dimensional system, gives the chaotic invariant sets are the results of computer simulation. The main contents of the paper are as follows: the first chapter mainly introduces some basic concepts of piecewise smooth systems and piecewise smooth dynamical systems. Research status. The second chapter mainly introduces the symbolic dynamics and topological theory. The third chapter mainly studies the necessary and sufficient conditions for a class of three-dimensional piecewise affine systems exist homoclinic orbit, a mathematical method for constructing chaotic system is presented, and on this basis, the structure of several chaotic systems, the computer simulation results is also given out. The fourth chapter introduces the necessary and sufficient conditions for a class of three-dimensional piecewise affine systems exist heteroclinic loops, and based on the use of topological horseshoe theory to prove the existence of chaotic invariant set, a mathematical method for constructing chaotic system is proposed. And on this basis, the structure of several chaotic systems, the computer simulation results are given related fifth chapter is existence of four-dimensional piecewise affine systems of singular bad. Necessary and sufficient conditions for the first study a class of four-dimensional piecewise affine systems exist double focus homoclinic orbit, a given The mathematical method of chaotic system construction, and based on the structure of a chaotic system, the simulation results are also given. The necessary and sufficient conditions related to study a class of four-dimensional piecewise affine systems bifocus heteroclinic loops exist, created a dual focus heteroclinic loop four-dimensional system, gives the chaotic invariant in the simulation results. The sixth chapter summarizes the whole work, and make the plan for the next step.

【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O415.5

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