An Essay On Dynamical Behaviors Of Stochastic Slow-fast Systems Driven By Non-gaussian Lévy Noise And Related Problems

Dynamical systems theory is an active and exciting area of modern mathematics which provides a powerful tool for studying systems evolving time and space.It is useful to unveil general and universal mechanisms underpinning the plethora of dynamical behaviors ob-served in nature.Furthermore,it also allows us to apply constructive ideas in different ways to understand dynamical systems in nature and technology along with applications ranging from biology to weather and climate.Lévy motions arise as appropriate models for non-Gaussian fluctuations with jumps in various complex systems varying from geophysics,finance,biophysics,and other disci-plines.They have independent and stationary increments as well as stochastically contin-uous sample paths with jumps.The?-stable Lévy motions?0<?2?,especially the Brownian motions??=2?,form special subclasses of Lévy motions.Solutions of SDEs driven by Lévy motions are Feller processes under certain conditions,where the infinitesi-mal generator of the corresponding semigroup can be constructed explicitly.Those solutions give rise to stochastic flows and hence generate random dynamical systems.This dissertation is to investigate the slow manifold of dynamical system?2.5?driven by?-stable Lévy process with??(1,2]in finite dimensional setting,and examine its approximation and structure.We analyze a detail structure of the large deviation principle for the slow variables{X_t^?}_?>0of slow-fast stochastic dynamical system?4.1?with?=?^?,?>1 in three different regimes.The layout of this thesis is as follows.We introduce in Chapter 1,the physical background and research progress on this the-sis.In Chapter 2,we recall some basic concepts in random dynamical systems and con-struct metric dynamical systems driven by Lévy processes with two-sided time.We intro-duce random invariant manifolds and some hypotheses for the slow-fast system?2.5?.In Chapter 3,we establish the existence of slow manifold and measure the rate of slow manifold attracts other dynamical orbits.We prove that as the scale parameter tends to zero,the slow manifold converges to the critical manifold in distribution.We present numerical results using examples from mathematical biology to corroborate our analytical results.In Chapter 4,we give some precise conditions for the slow-fast system,and describe the Cauchy problem?4.13?satisfied by{U^?}_?>0and introduce the limit Hamiltonian H⁰that has different forms in the three regimes depending on?:supercritical case for?>2,critical case for?=2 and subcritical case for?<2.In Chapter 5,we derive the comparison principle and present the convergence result for solutions of the Cauchy problem?4.13?with H^?to the unique viscosity solution of the Cauchy problem?4.33?with H⁰.The large deviation principle for the slow variables{X_t^?}_?>0is presented.In the end,some problems for further research on the related topics are given.