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

确定性和随机扰动下动力系统的长时间和瞬态行为

• 批准号: RGPIN-2020-04451
• 负责人: Yi, Yingfei
• 金额: $ 2.7万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739239
• 关键词: Long; time; transient; behaviors; dynamical

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.

该研究计划旨在研究连续或离散动态系统在自治或非自治、随机或随机性质的大范围扰动下的长时间和暂态动态以及复杂性。它由四个独立但又相互关联的项目组成：一、哈密顿系统中的准周期运动；二、噪声对耗散系统动力学的影响；三、随机系统中的准平稳动力学；以及四、马尔可夫随机网络中的同步与去同步。项目I研究了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拟周期运动的存在性和线性稳定性。这些都是哈密顿动力学中的开放领域，其研究需要KAM理论的新发展。项目II涉及在自治和非自治环境中白噪声扰动的耗散常微分方程组中关于基本动力学问题的平稳度量的持续研究，例如随机稳定性、收敛、集中度和分叉。这项研究有望受益于我以前关于这一主题的工作中介绍的水平集方法，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退化，还需要开发新的技术。项目III研究空间离散和连续随机系统中的暂态随机动力学，重点研究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。与已有研究较多的亚稳定随机暂态现象相比，准平稳分布的性质和机理还远未得到很好的理解。项目四研究离散随机网络的马尔可夫扰动中的同步和去同步现象，即既具有内在不确定性又具有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出现在生物应用中的新的随机动力系统，它的研究将把各种动态和随机对象联系在一起。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，这些项目的成果有望极大地丰富动力系统理论以及加拿大在确定性、随机和随机系统中产生的扰动、长时间和瞬时动力学方面的科学研究领域。它们应该在科学和工程的更广泛领域有重要的应用，包括天体和统计力学、环境科学、人口和细胞生物学、固体物理和生物化学等。该计划还将为HQP培训提供一个极好的机会。