Asymptotic Behavior Of Non-autonomous FitzHugh-Nagumo Lattice System With Multiplicative White Noise

In this thesis,we mainly studied the existence of a random exponential attractor for non-autonomous FitzHugh-Nagumo lattice system with multiplicative white noise and the existence,structure and stability of random attractors for non-autonomous timedelay FitzHuge-Nagumo lattice systems with multiplicative white noise.The content of the thesis is divided into the following three chapters:In the first chapter,we first present the development background of related issues,then we introduce the main content of this thesis and the basic concepts related to this thesis.In the second chapter,we consider the existence of a random exponential attractor for non-autonomous FitzHugh-Nagumo lattice system with multiplicative white noise.Firstly,we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle defined on a weighted space of infinite sequences.Secondly,we transferred the stochastic FitzHugh-Nagumo lattice system with multiplicative white noise into a random FitzHugh-Nagumo lattice system with random coefficients and without white noise by the Ornstein-Uhlenbeck process.Thirdly,we estimated the“bound and tail”of solutions of the random system,and decomposed the difference between two solutions into a sum of two parts,and estimated the boundedness of the norm of each part and the expectations of some random variables.Finally,we obtained the existence of a random exponential attractor for the considered system.In the third chapter,we consider the existence,structure and stability of random attractors for non-autonomous delay FitzHugh-Nagumo lattice systems with multiplicative white noise.Firstly,we introduce the sufficient conditions for the existence of random attractors for non-autonomous stochastic delay lattice systems.Secondly,we prove the existence of random absorbing sets for non-autonomous delay FitzHuge-Nagumo lattice systems,then we estimated the tail of the solution,and hence we obtained that the system has a unique tempered random attractor,and we proved that the random attractor of the system is a single point set,and therefore,the system has a unique tempered complete quasi-solution which exponentially pullback attracts all the solutions starting from a tempered random set.Finally,we prove the convergence of the tempered complete quasi-solution when time delay tends to zero.