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

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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.