Deterministic and Stochastic Perturbations of Dynamical Systems

围绕动力系统的扰动理论，研究计划包括三个独立但相关的项目：I。近可积保守系统共振区的准周期运动; II。几乎自守动力学作为多频振荡系统中的不稳定性; III。耗散系统的随机稳定性。项目一，二关注自然界中的多频现象的研究，包括保守或耗散的生物，电气和机械系统中产生的现象。大量的实验和计算机模拟表明，多频系统可以产生规则的、复杂的多相振荡的动力学结果。但是，这种规律性和复杂性的机制还远未得到很好的理解，特别是当多频振荡发生在共振区或所考虑的系统是不可积的。项目三是一个系统的研究随机稳定性分布的角度来看，在本地和全球水平，为白色噪声扰动有限维，耗散系统。物理系统经常受到来自其周围环境或其内在内部不确定性的噪声扰动。分析噪声扰动对这些系统动态特性的影响成为建模和分析的基本问题，但从分布角度分析这种噪声影响的理论基础需要进一步发展。在这些项目中，将开发各种数学理论，以基本理解由于多频率或噪声扰动的相互作用而导致的广泛的动力系统的复杂性，复杂性和稳定性。这些项目的结果预计将大大丰富微分方程动力学理论以及加拿大多频振荡和噪声影响方面的科学研究领域，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。它们还将在科学和工程的更广泛领域中有重要的应用，包括经典和天体力学、环境科学、材料科学、人口和细胞生物学、固态物理学和生物化学等。

Centered around perturbation theory of dynamical systems, the research program consists of three separate but related projects: I. Quasi-periodic motions in the resonance zone of a nearly integrable conservative system; II. Almost automorphic dynamics as intermittency in multi-frequency oscillatory systems; III. Stochastic stability of dissipative systems. Projects I, II concern the study of multi-frequency phenomena in nature including those arising in either conservative or dissipative biological, electrical, and mechanical systems. It is predicted by many experiments and computer simulations that systems involving multi-frequency can produce both regular and complex dynamical outcomes oscillating in multi-phases. But the mechanisms of such regularity and complexity are far from being well-understood, especially when either the multi-frequency oscillations occur in the resonance zones or systems under consideration are less integrable. Project III is a systematic study of stochastic stability from distribution perspectives at both local and global levels, for white noise perturbed finite dimensional, dissipative systems. Physical systems are often subject to noise perturbations either from their surrounding environments or from their intrinsic internal uncertainties. Analyzing the impact of noise perturbations on the dynamics of these systems then becomes a fundamental issue with respect to both modeling and analysis, but a theoretical foundation for analyzing such noise impacts from a distribution perspective needs to be further developed.******In these projects, various mathematical theories are to be developed toward making fundamental understandings of regularities, complexities, and stabilities of a broad range of dynamical systems due to either the interaction of multi-frequencies or noise perturbations. Results from these projects are expected to substantially enrich the theory of dynamics of differential equations  as well as scientific research fields in Canada with respect to multi-frequency oscillations and the impact of noises, by introducing new theories, new areas of research, new methodology, as well as a number of new dynamical phenomena. They will also have significant applications to a broader range of areas in science and engineering including classical and celestial mechanics, environmental science, material science, population and cell biology, solid-state physics, and bio-chemistry, etc.********************