Qualitative Analysis Of Several Nonlinear Dynamical Systems

With the rapid development of modem science and technology,there are a lot of nonlinear problems in various fields of science and technology and disciplines,espe-cially physics,mathematics,chemistry,biology,sociology and other disciplines,and applications more widely,so how to solve these nonlinear problems becomes more and more important,which can be described by some nonlinear dynamical systems.How-ever,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 differential equations.In a certain parameter conditions and boundary conditions,nonlinear dynamical system will show a complilated behavior,so it is necessary and research valuable to inves-tigate the dynamic behavior of nonlinear dynamical systems under certain boundary condition and parameter condition.In this paper,we deeply discuss the stability and bifurcations of several kinds of nonlinear partial differential equations and the full text is divided into five chapters.The first chapter is the introduction,which briefly introduce the present situation of the stability and bifurcations of the nonlinear partial differential equation and the main work of this article and the structure arrangement.In the second chapter,bifurcation and stability of two kinds of constant stationary solutions to a non-reversible amplitude equation are investigated on a bounded domain with Neumann boundary conditions by using Mathematics and Matlab.The perturbed term destroyed the reversibility of amplitude equation and the corresponding equilibri-um changed from a center into a focus while the other two equilibria keep unchanged.In the third chapter,we study a reaction-diffusion equations with cross-diffusion terms by using the multi-scale method.We show that cross-diffusion is the necessary ingredient for pattern formation through a linear stability analysis,and perform a weak-ly nonlinear analysis to predict the amplitude and the form of the pattern near marginal stability.The Ginzburg-Landau amplitude equation and the corresponding exact solu-tions and bifurcation diagrams are given using Matlab and AUTO.In the fourth chapter,we consider continuous-wave states of a nonlinear coupled system.Both symmetric and asymmetric continuous-wave states are found,and the symmetry-breaking bifurcation,which gives rise to asymmetric continuous-wave back-grounds,is identified.The stability of solutions is also investigated in this section.By discussing the modulation stability in symmetric and asymmetric backgrounds,we can not obtain homoclinic orbits at symmetry-breaking bifurcation.Finally,in the fifth chapter we summarize the full thesis and give some conclusions and outlook.