Long time and transient behaviors of dynamical systems under deterministic and random perturbations

提出的研究计划旨在研究长时间和瞬态动力学以及在自主或非自主，或随机或随机性质的广泛扰动下连续或离散动力系统的复杂性。它由四个独立但相关的项目组成：1 .哈密顿系统中的准周期运动；2。噪声对耗散系统动力学的影响3。随机系统的准平稳动力学；四、马尔可夫随机网络中的同步与去同步。项目一关注拟周期运动在近可积多尺度哈密顿系统和弱耦合有限光滑哈密顿网络中的存在性和线性稳定性。这些都是哈密顿动力学的广阔领域，其研究需要对KAM理论进行新的发展。项目二涉及在自主和非自主环境下的白噪声扰动、耗散ODE系统中，关于随机稳定性、收敛、集中和分岔等基本动力学问题的平稳措施的持续研究。本研究预计将受益于我之前关于该主题的作品中引入的水平集方法，但考虑到自治情况下吸引子的复杂性和非自治情况下噪声的简并性，还需要开发新的技术。项目三关注空间离散和连续随机系统中的瞬态随机动力学研究，重点关注在许多应用中代表物理相关平稳的准平稳分布的存在和行为。与研究较多的随机暂态现象相比，拟平稳分布的性质和机制还远远没有得到很好的认识。项目IV涉及离散随机网络马尔可夫摄动中的同步和反同步现象的分析，即具有内在不确定性和外在随机性的离散时间、离散状态网络。这些网络在生物学应用中形成了一类新的随机动力系统，它的研究将把各种动态和随机对象联系在一起。

The proposed research program aims at studying both long time and transient dynamics and complexities of continuous or discrete dynamical systems under a broad range of perturbations of either autonomous, or non-autonomous, or random, or stochastic natures. It consists of four separate but related projects: I. Quasi-periodic motions in Hamiltonian systems; II. Noise impacts on dynamics of dissipative systems; III. Quasi-stationary dynamics in stochastic systems; and IV. Synchronization and de-synchronization in Markov random networks. Project I concerns the existence and linear stability of quasi-periodic motions arising in nearly integrable, multi-scale Hamiltonian systems and weakly coupled, finitely smooth Hamiltonian networks. These are wide-open areas in Hamiltonian dynamics whose study requires a novel development of the KAM theory. Project II concerns a continuing study on stationary measures, with respect to fundamental dynamics issues such as stochastic stability, convergence, concentration, and bifurcation, in white noise perturbed, dissipative ODE systems in both autonomous and non-autonomous settings. This study is expected to benefit from the level sets method introduced in my previous works on the subject but new techniques also need to be developed given the complexity of the attractors in the autonomous case and the degeneracy of noises in the non-autonomous case. Project III concerns the investigation of transient stochastic dynamics in both spatially discrete and continuous stochastic systems, focusing on the existence and behaviors of quasi-stationary distributions which represent physically relevant stationarities in many applications. Comparing with the well studied stochastic transient phenomenon due to meta-stability, quasi-stationary distributions are far from being well understood in terms of their properties and mechanisms. Project IV concerns the analysis of synchronization and de-synchronization phenomena in Markov perturbations of discrete random networks, i.e., discrete-time, discrete-state networks with both intrinsic uncertainties and extrinsic randomness. These networks form a new class of random dynamical systems arising in biological applications, whose study will connect various dynamical and stochastic objects together. Results from these projects are expected to substantially enrich the theory of dynamical systems as well as scientific research fields in Canada with respect to pertrubative, long time and transient dynamics arising in deterministic, random, and stochastic systems, by introducing new theories, new areas of research, new methodology, as well as a number of new dynamical phenomena. They should have significant applications to a broader range of areas in science and engineering including celestial and statistical mechanics, environmental science, population and cell biology, solid-state physics, and bio-chemistry etc. The program will also provide an excellent opportunity for HQP training.