Dynamics Analysis Of Several Nonlinear Dynamical Systems

As we know, with the rapid development of science and technology, on the one hand, there are a lot of nonlinear problems which can be described by many nonlin-ear dynamic systems, those systems play an important role in mathematics, physics, chemistry, biology, engineering, economic, etc. On the other hand, there are great achievements in the investigation of nonlinear partial differential equation. A lot of important science and technology issues can be described by nonlinear partial differ-ential equations. Under the certain condition of parameters, the dynamics of nonlinear dynamical system can show different movements which has some unexpected dynam-ics, so, dynamical behaviors of the nonlinear dynamical systems that under a certain condition of parameter is necessary and meaningful. In fact, many methods are applied to study the stability and bifurcations of nonlinear partial differential equations.In this paper, using the unstable short wave method, multi-scale analysis methods and Euler discrete method, we study the stability and bifurcations of several kinds of differential equations, the full text is divided into six chapters.The first chapter is the introduction. The present situation of the stability and bifurcations of the differential equation of system and the main work of this article and the structure arrangement.The second chapter introduces three research methods used in this article:method of the unstable short wave method, multi-scale analysis method and Euler discrete method.The third chapter, in the Lagrangian coordinates, using the method of instability short wave, we study stability of the undercurrent of equatorial waves on/plane.The fourth chapter studies a class of generalized Swift-Hohenberg equations and and we obtain the corresponding snaking bifurcations by AUTO.The fifth chapter studies a discrete-time predator-prey system of Leslie type with generalized Holling type Ⅲ functional response.Finally, in the sixth chapter, we summarize the full thesis and point out the further research work.