离散李雅普诺夫泛函作用下的非自治系统与指数分离性

The proposed research will focus on the nonautonomous systems with Discrete Lyapunov Functional, as well as its exponentially separating property. More precisely, in the general framework of infinite dimensional skew-product semiflows, we will construct the k-dimensional exponential separation with respect to the k-cones.The equivalent relation between Discrete Lyapunov Functional and the existence of infinitely many nested k-cones will be studied via the approach of constructing such exponential separation. The theory of invariant manifolds and foliations associated with k-dimensional exponential separation are also investigated. Based on these tools, by considering the persistence of the nested k-cones under smooth perturbation, we will make an attempt to investigate the lifting property of the minimal (limit-) sets, the finite-dimensional imbedding structure and such structural stability under C1-pertubations. The theory will also be, in this project, applied to various non-gradient differential equations, such as 1-D nonautonomous parabolic equations (under periodic or separated boundary conditions), nonautonomous (delayed) feedback cyclic equations and reaction-diffusion equations on the thin domains, etc., both as testing and a guiding tool for trying to solve the structure of invariant sets,its structural stability and the new phenomena of almost automorphy in specific systems under consideration.

本项目研究离散Lyapunov泛函作用下的非自治系统的性态及其指数分离性。在抽象无穷维斜积半流的框架下，研究k秩锥诱导的斜积系统对应的k 维指数分离性质，研究相应指数分离性下的不变流形存在性及叶层理论。建立离散Lyapunov 泛函作用与嵌套k秩锥簇存在性的等价关系，并考察嵌套锥簇关于斜积半流光滑扰动的保持，进而研究离散Lyapunov 泛函作用下的斜积半流极小集（及极限集）的提升性质，维数嵌入性质及其相应的C1-扰动结构稳定性。利用此统一的观点来处理包括一维非自治抛物方程（周期边条或分离边条）、循环反馈的非自治（时滞）微分方程、薄域上的非自治反应扩散方程等多类非梯度类方程的不变集结构、扰动稳定性以及新的几乎自守集的存在性。

本项目深入研究了离散Lyapunov泛函作用下的非自治系统的动力学性态及其指数分离性，揭示了其与动力系统锥不变性的密切关联，并通过其获得了多类微分方程的不变集特性与结构稳定性。相关的内容主要包含以下几个方面：(1) 在无穷维系统框架下，以离散Lyapunov泛函生成的不变锥为背景，揭示了动力系统的锥不变性与随机指数分离性的拟等价性及其与动力系统乘积遍历理论的紧密关联；证明了高秩不变锥下的无穷维随机系统Krein-Rutmann 型定理。(2) 对高秩不变锥下的无穷维非线性系统，建立极限集的伪有序原理并发现同宿现象，进而证明无穷维系统的Poincare-Bendixson定理。(3) 研究了非紧拓扑群作用下的单调斜积半流稳定极小集的群对称及群单调性态, 证明了无界对称区域上非自治反应扩散方程稳定整解的空间对称性。(4) 利用离散Lyapunov泛函研究几乎周期外频驱动的一维抛物方程周期边值问题的动力学。在非线性项反射对称情形下，完整刻划了极限集结构和遍历性，发现了与单频驱动的系统本质不同的几乎自守现象。(5) 通过离散Lyapunov泛函建立循环反馈微分方程的Floquet丛理论，证明了相应系统的不变流形横截相交性，显示了其结构稳定性。