非一致双曲测度链动力系统稳定性的若干问题

The theory of nonuniform hyperbolicity describes the theory of dynamical systems whose Lyapunov exponents are not zero,and lies in the heart of the modern theory of dynamical systems. Dynamical systems on measure chains is a new research field and has a potential application value. However, there are no results to consider the stability of nonuniformly hyperbolic dynamical systems on measure chains. In this project, we systematically explore the problems of the stability of nonuniformly hyperbolic dynamical systems on measure chains with the help of nonuniform exponential dichotomies and nonuniform exponential trichotomies..In the case of linear perturbations, we establish the robustness and the structural stability of nonuniformly hyperbolic dynamical systems on measure chains in Banach spaces assuming that the corresponding linearized systems admit a nonuniform exponential dichotomy or trichotomy. In the case of nonlinear perturbations,we investigate the H?lder continuity and the smoothness of the topological linearization and explore the existence of stable invariant manifolds which provide the geometric structures for describing and understanding the qualitative behavior of nonlinear dynamical systems. In particular, we focus on the effects and the.internal rules of the different measure chains for the stability problems on the above discusses and would reveal the essential difference between continuous dynamical systems and discrete dynamical systems. Therefore, this research project not only enriches the theory of nonuniformly hyperbolic dynamical systems on measure chains, but also may promote the development of the theory of nonuniform hyperbolicity and its related discipline.

非一致双曲性理论刻画了具有非零Lyapunov指数的动力系统，是当今动力系统研究中的前沿课题之一。测度链动力系统是动力系统中一个新兴的研究领域，具有潜在的应用前景。而目前把二者结合起来研究非一致双曲测度链动力系统稳定性问题的结果还不多见，有很多问题尚待解决。本项目拟以测度链上非一致指数型二分性和三分性为上述两种理论结合的切入点，系统地探讨非一致双曲测度链动力系统的稳定性问题。建立非一致双曲测度链动力系统在线性扰动下的鲁棒性与结构稳定性，研究其在非线性扰动下拓扑线性化的H?lder连续性和光滑性，详细地探讨在其稳定性中具有重要几何意义的不变流形理论。特别将着重阐明不同测度链的选取对上述动力系统稳定性问题研究的影响及其内在规律，从而有助于揭示连续动力系统和离散动力系统的本质差异。因此本项目的研究不仅可以丰富非一致双曲测度链动力系统自身的理论，而且也可能推动非一致双曲性理论及其相关学科的发展。

非一致双曲性理论和测度链动力系统是当代动力系统研究中的重要问题。本项目将二者结合起来对非一致双曲测度链动力系统稳定性问题进行了深入研究。我们首先给出了测度链上非一致指数型二分性与非一致指数型三分性的概念，利用李雅普诺夫指数，李雅普诺夫函数，函数对空间的允许性，建立其存在的判别准则。基于测度链上非一致指数型二分性与非一致指数型三分性，系统地探讨了非一致双曲测度链动力系统的稳定性问题，如鲁棒性、结构稳定性、拓扑线性化定理、不变流形存在性定理等。特别是分析了不同测度链的选取对上述动力系统稳定性问题研究的影响及其内在规律。本项目通过三年的研究工作，已基本上完成项目计划内容。在本项目的执行过程中，共发表学术论文9篇，其中8篇被SCI收录。