Dynamical Behavior Of FitzHugh-Nagumo System Driven By Nonlinear Noise

FitzHugh-Nagumo system is a coupled nonlinear differential equations presented by FitzHugh and Nagumo,which describing the initiation and transmission of action potentials in neurons,and is an important mathematical model.This thesis deals with the Wong-Zakai approximations and random attractors for stochastic FitzHugh-Nagumo system with a nonlinear noise.We first prove the existence of a pullback random attractor for the approximate equation under much weaker conditions than the original stochastic equation on bounded domains.Then,when the stochastic FitzHugh-Nagumo system is driven by a linear multiplicative noise,we establish the convergence of solutions of Wong-Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation tends to zero.Finally,we study dynamical behavior of FitzHugh-Nagumo system on unbounded domains.This thesis is organized as follows:In Chapter 1,we introduce the research background of stochastic FitzHugh-Nagumo system,and briefly describe the main work of this thesis.In Chapter 2,we present some definitions and theorems of random attractors for random dynamical systems.In Chapter 3,we study Wong-Zakai approximations of the stochastic FitzHugh-Nagumo system on bounded domains,then prove the existence of the random attractor and the upper semicontinuity of the stochastic FitzHugh-Nagumo system driven by linear multiplicative noise.In Chapter 4,we consider the Wong-Zakai approximations and dynamical behavior of FitzHugh-Nagumo system on unbounded domains.In Chapter 5,we summarize the main results and propose some problems for future research.