Study On The Stability Of Several Nonlinear Dynamical Systems

With the rapid development of modern science and technology,there are a lot of nonlinear problems in various fields of science and technology and disciplines,includ-ing physics,mathematics,chemistry,biology,sociology and other disciplines,which is widely applied in many subjects,so how to solve these nonlinear problems becomes more and more important,actually,most of them can be described by some nonlinear dynamical systems.However,using nonlinear partial differential equations to describe the problems,we can fully take into account the impact of space,time and time delay,thus more accurate reflection of the actual situation.Many important natural sciences and some technical problems can be considered as the subject of nonlinear partial differ-ential equations.In a certain parameter conditions and boundary conditions,nonlinear dynamical system will show a compliated behavior,so it is necessary and research valu-able to investigate the dynamics of nonlinear dynamical systems under certain boundary condition and parameter condition.In this paper,the stability of several nonlinear partial differential equations is s-tudied.It is divided into four chapters.The chapter 1 is the introduction.The stability of the steady-state solution of several nonlinear partial differential and Turing on stability of nonlinear system are bricfly given,including current development status and composition of the whole thesis.In chapter 2,the global asymptotic stability of equilibrium solutions for Holling type equations with disturbed diffusion terms is studied by using Lyapmnov function-al.This perturbation breaks the equilibrium state of system and turns the stable static solution into a periodic solution.In chapter 3,by using multi-scale analysis a class of reaction diffusion equations with cross dissipation terms is studied.The linear stability analysis is used to investi-gate the nonlinear dissipative system with a cross dissipative term which is an essential component of the pattern.This will produce the normal form Ginzburg-Landau ampli-tude equation.In chapter 4 the global asymptotic stability of the equilibrium solution of predator and prey models with disturbed diffusion terms is investigated by using a new Lyapm-nov functional.Finally,we summarize the main results of the thesis and the future topics in the fifth chapter.