Deterministic and Stochastic Perturbations of Dynamical Systems

本研究以动力系统的摄动理论为中心，由三个相互独立但又相互关联的项目组成：一、近可积保守系统共振区中的准周期运动；二、多频振荡系统中具有间歇性的几乎自同构动力学；三、耗散系统的随机稳定性。项目I、II涉及研究自然界中的多频率现象，包括那些在保守的或耗散的生物、电气和机械系统中出现的现象。许多实验和计算机模拟预测，涉及多个频率的系统可以产生规则的和复杂的多相振荡的动力学结果。但这种规律性和复杂性的机制还远未被很好地理解，特别是当发生在共振区的多频振荡或所考虑的系统较不可积性时。项目III是从局部和全局分布的角度对白噪声扰动的有限维耗散系统的随机稳定性进行的系统研究。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力学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还有待进一步发展。在这些项目中，将发展各种数学理论，以基本理解由于多个频率的相互作用或噪声扰动而导致的广泛的动力系统的规律性、复杂性和稳定性。这些项目的成果预计将通过引入新理论、新研究领域、新方法以及一些新的动力学现象，大大丰富微分方程动力学理论以及加拿大关于多频振荡和噪声影响的科学研究领域。它们还将在更广泛的科学和工程领域有重要的应用，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、固体物理和生物化学等。

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.********************