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

拟议的研究计划的目的是研究长期和瞬态动态和复杂性的连续或离散动力系统在广泛的扰动下，无论是自治的，或非自治的，或随机的，或随机性质。它包括四个独立但相关的项目：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; III.随机系统中的准平稳动力学;以及IV.马尔可夫随机网络中的同步与去同步。项目一涉及的存在性和线性稳定性的准周期运动中产生的几乎可积，多尺度哈密顿系统和弱耦合，光滑的哈密顿网络。这些都是在哈密顿动力学的研究需要一个新的发展的KAM理论的广阔的开放领域。项目二是关于静态措施的持续研究，在白色噪声扰动，耗散ODE系统在自治和非自治设置的基本动力学问题，如随机稳定性，收敛性，浓度和分歧。这项研究预计将受益于水平集方法在我以前的作品中介绍的主题，但新的技术也需要开发的复杂性的吸引子在自治的情况下和退化的噪声在非自治的情况下。项目III关注空间离散和连续随机系统中的瞬态随机动力学的研究，重点关注在许多应用中代表物理相关平稳性的准平稳分布的存在和行为。与亚稳定性引起的随机瞬态现象相比，准平稳分布的性质和机理还远未得到很好的理解。项目四涉及离散随机网络的马尔可夫扰动中的同步和去同步现象的分析，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。这些网络形成了一类新的随机动力系统，在生物学应用中出现，其研究将各种动态和随机对象连接在一起。从这些项目的结果预计将大大丰富动力系统的理论，以及科学研究领域在加拿大方面的扰动，长时间和瞬态动力学所产生的确定性，随机和随机系统，通过引入新的理论，新的研究领域，新的方法，以及一些新的动力学现象。他们应该有显着的应用范围更广的领域，包括天体和统计力学，环境科学，人口和细胞生物学，固体物理学和生物化学等科学和工程该计划也将提供一个极好的机会，为HQP培训。

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.