Bifurcation And Center Conditions For Several Kind Of Differertial Autonomous Systems

This thesis is devoted to the problems of integral conditions, center- focus determination and bifurcation of limit cycles at the origin and the infinity of planar polynomial differential system. It is composed of four chapters.In chapter 1, it is introduced and summarized about the historical background and the present progress of problems about center-focus determination and bifurcation of limit cycles of planar polynomial differential system. At the same time, the main work of this paper is concluded.In chapter 2, it is studied singular quantities and center conditions for a class of quintic polynomial system. First, singular quantities at the origin of the system are discussed, then by means of the transformation, the study of the infinity can be changed into the origin of the system, and, with Mathematical, the first 6 singular quantities at the origin and the first 12 singular quantities at the infinity are deduced. At the same time, the center conditions of the origin and the infinity are derived.In chapter 3, the center conditions and bifurcation of limit cycles for a class quasi cubic systems are investigated. First the twenty-first singular point are computed and conditions for origin to be a center are deduced as well, then a system that bifurcations four limit cycles at the origin or the infinity are constructed.In chapter 4, as social progress and ceaseless scientific research, a large number of nonlinear mathematical models appear in actual engineering and various branches of natural sciences, even, in the domain of social sciences, which waiting for being studied deeply by science workers. The explicit solution expressions of these nonlinear issues are difficult obtained, compare to the case of linear equations. it becomes an important topic to study kinds of finity travelling waves which including solitary waves, a lot of techniques and methods have been developed such as the inverse scattering method, Darboux transformation method, Hirota bilinear method, tanh method and so on. However, except determining the explicit solutions under the given conditions, these methods can't give the integrated explanation about the relationship between pa-rameters and the existence of peculiar finity solutions. Latest researches indicate the theory of bifurcation method and qualitative analysis of dynamical system can make up these deficiencies, even, by using the theoretics of dynamical system, we can get much deep, nonlinear simultaneous Schr(o|¨)dinger equation has been studied in the light of the theory of dynamical systems and the theory of bifurcation. The existence of smooth solitary wave and kink and periodic wave solutions have been proved. Conditions sufficient for the existence of smooth solitary wave solutions and kink wave solutions and periodic wave solutions under different parameters have been given, the method of all exact explicit formula of above solutions is also given.