Three-component Reversible Gray - Scott System And Random Attractor Of Lattice Dynamic System

It is well known that attractor is the appropriate tool to study the long timeasymptotic behaviors of the solutions of infinite dimensional dynamical systems.Among the reaction-diffusion systems, the three-component reversible Gray-Scottsystem is a vita model to describe the scheme of two reversible chemical orbiochemical reactions. In this doctoral dissertation, we mainly consider the existenceof global attractors for the three-component reversible Gray-Scott system and thelattice dynamical systems when perturbed by the fractional Brownian noise and Levynoise. These results not only can enrich the contents in the research of the dynamicalsystems, but also can provide new insights to the related research areas, such as,chemistry, biochemistry, biology, science of material, etc.The first section consists of the first two chapters. In chapter1, we firstintroduce the background and status of these aspects of research, as well as thecontents and significance of this study. Chapter2recalls some preliminary results.In section2, we mainly discuss the existence of global attractors for thethree-component reversible Gray-Scott system. Chapter3considers the uniformattractor for the non-autonomous three-component reversible Gray-Scott systemwith Dirichlet boundary condition on a bounded domain and the pullback attractoron unbounded domains. Chapter4obtains the existence of random attractors for thestochastic three-component reversible Gray-Scott system on bounded domain andunbounded domains, respectively.Section3comprises chapters5. We respectively establish the existence of globalrandom attractors of the stochastic three-component reversible Gray-Scott system,the first-order stochastic lattice dynamical system with fractional Brownian noise and Levy noise.