无穷维双曲系统的混沌动力学研究

Chaos, as a description of the complex of dynamical systems, has attracted the great interest of researchers due to its wide existence in nature. In this project, we will study the complex chaotic dynamics of infinite-dimensional dynamical systems governed by hyperbolic partial differential equations (PDEs) from nonlinear boundary condition and state space, respectively. The main contents include: (1) Investigate chaos in the sense of exponential growth of total variation with time for the systems governed by one-dimensional linear hyperbolic PDEs with both boundary conditions being implicit boundary conditions by using of characteristics and the related theories of discrete dynamical systems; (2) Based on the research of the one-dimensional chaotic system, the chaotic dynamics of two-dimensional linear hyperbolic PDEs are further studied; (3) Show the existence of chaos for hyperbolic PDEs on the space of analytic functions and provide Devaney chaotic and hypercyclic criteria for the corresponding solution semigroups by using of the related theories of topological dynamical systems. The results obtained further help to explore the intrinsic relationship between chaos and related dynamics of hyperbolic systems and state space. Thereby, these research results will further enrich and develop the chaos theory of infinite-dimensional dynamical systems, and reveal the mechanism of chaos in hyperbolic PDEs.

混沌作为动力系统复杂性的一种刻画，普遍存在于自然界中，其理论和应用研究引起了广泛的关注。本项目从非线性边界条件和状态空间两方面因素来研究由双曲型偏微分方程诱导的无穷维动力系统的复杂混沌动力学特性。本项目拟研究的内容包括:（1）基于偏微分方程和离散动力系统相关理论，探究两端都具有隐式边界条件的一维线性双曲系统在总变差指数增长意义下混沌的存在性问题:（2）在一维线性双曲系统混沌的基础之上，进一步研究具有非线性边界条件的二维线性双曲系统的混沌动力学:（3）利用拓扑动力系统相关理论，证明双曲系统在具有一定增长控制的解析函数空间上混沌的存在性，给出双曲系统解半群的Devaney混沌和超循环性判据，进而延伸探讨双曲系统混沌及相关动力学行为与所选状态空间的内在联系。研究成果将进一步丰富和发展无穷维动力系统的混沌理论，且揭示双曲型偏微分方程产生混沌的机理。

混沌作为动力系统复杂性的一种刻画，普遍存在于自然界中，其理论和应用研究引起了广泛的关注。本项目主要研究了具有非线性边界条件的双曲型偏微分方程诱导的无穷维动力系统的复杂混沌动力学。主要在如下四方面取得了系列研究成果：1.基于偏微分方程和离散动力系统相关理论，证明了两端都具有隐式边界条件的一维线性双曲系统在总变差指数增长意义下混沌的存在性；2.在一维线性双曲系统混沌的基础之上，给出了具有非线性边界条件的二维线性双曲系统的混沌动力学；3.利用回归排斥子和异宿环理论，严格证明了具有线性和超线性边界条件的三维线性双曲系统的混沌动力学；4.给出了一类偏微分方程混沌系统的观测器设计和稳定性。这些结果都给出了仿真实例验证。研究成果不仅有助于丰富和发展偏微分方程的混沌理论，而且有助于进一步揭示双曲型偏微分方程产生混沌的机理，先后发表在Chaos、Journal of Mathematical Analysis and Applications、Nonlinear Dynamics等国际重要期刊上。