Deterministic and Stochastic Perturbations of Dynamical Systems

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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.