Morse decomposition for random dynamical systems
July 05, 2019 16:30-18:00
The Morse decomposition theorem states that a compact invariant set of a given flow can be decomposed into finite invariant compact subsets and connecting orbits between them, which is helpful for us to study the inner structure of compact invariant sets. When dynamical systems are randomly perturbed, by real or white noise, we show that for finite and infinite dimensional random dynamical systems, we have the random Morse decomposition; we also construct Lyapunov function for the decomposition. For deterministic systems, we introduce the concept of natural order to study the relative stability of Morse sets by the stochastic perturbation method. We also investigate the stochastic stability of Morse (invariant) sets under general white noise perturbations when the intensity of noise converges to zero.