Limit Cycles And Critical Period Bifurcations Of Some Classes Of Planar And Spatial Differential Systems

Using the qualitative theory and bifurcation theory of differential equations.We study the center and isochronous center,limit cycle and local critical period bifurcation problem for several kinds of differential systems in this thesis,which is composed of five chapters.The first chapter discusses the research background and research status of limit cycle and critical period bifurcation,center and isochronous center of smooth and piecewise smooth differential systems,and introduces the specific contents and innovations of each chapter.In Chapter 2,the isochronous center conditions and the number of critical period bifurcation for a class of quintic Z₂ equivariant real systems are studied.A set of necessary and sufficient conditions for the equilibrium point of the system to become an isochronous center are obtained,and it is proved that the system can have 8 local critical period bifurcation at the two symmetric singularities.In Chapter 3,the center and limit cycle of bifurcation for a class of three-dimensional cubic systems with two symmetric singularities are studied.The necessary and sufficient conditions for the two singularities to become the 8th order fine focus and the center are proved respectively,and it is obtained that there are at least 16 small amplitude limit cycles around the two symmetric singularities.In Chapter 4,the limit cycle and local critical period bifurcation for a class of quartic piecewise smooth differential systems are studied.It is proved that there are at least 18 limit cycles and 4 local critical period bifurcation around the two symmetric singularities.The fifth chapter summarizes and prospect the work of this thesis.