On Asymptotic Phenomena Of Dynamical Systems Driven By Lévy Processes

In this thesis, we study synchronization phenomena and invariant stable and un-stable manifolds for dynamical systems driven by non-Gaussian Levy noises.First we investigate synchronization of systems under additive as well as multi-plicative Levy noises. In particular, we consider coupled SDE systems in Rd, driven by Levy motion and where a, b, c, d are constants, Lti,i= 1,2,3,4 are independent two-sided Levy mo-tions, continuous functions f, g satisfy the one-sided dissipative Lipschitz conditions, and u* is the sum of stationary solutions of two Langevin equations.After discussing the stationary orbits and random attractors, a synchronization phenomenon is shown to occur. The synchronization result implies that coupled dy-namical systems share a dynamical feature in some asymptotic sense. This further develops the results of Caraballo and Kloeden.Random invariant manifolds are geometric objects useful for understanding com-plex dynamics under stochastic influences. We consider a Marcus systemUnder suitable assumptions on matrixes S, U and nonlinear terms f, g, invari-ant stable and unstable manifolds for random dynamical systems generated by above system are considered. The existing work in this area is for stochastic differential equations driven by noises that are continuous in time. Moreover, when the noise in- tensity is small, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold.Finally, the slow manifold of the following system is considered and an asymptotic result forε↓0+ is shown.