Limit Cycles And Bifurcations Of Critical Periods For Several Classes Of Differential Models

The paper is divided into five chapters.such problems of limit cycles,periodic traveling wave solutions and local critical periodic bifurcations for several types of polynomial differential models are mainly discussed.In the first Chapter,the research background and current situation of the above three questions are mainly expounded,and the main researches of each chapter are introduced briefly.In Chapter two,the limit cycles and local critical period bifurcations of a class of cubic systems with the item ofy³ are considered,and the origin to be the sixth order fine focus and the third order fine center are studied,the conclusion of 6 limit cycles and 3 critical period bifurcations are obtained and the correlation proof are given.In Chapter three,the problem of periodic traveling wave solutions of a class of biological invasion model with density dependence on migration and Allee effect for nonlinear partial differential equations of advection-diffussion-reaction type are studied,the traveling wave equation can be received by traveling wave transformation,through the computer algebra Mathematica the first several focus values can be obtained,by investigating its Hopf bifurcation,two limit cycles are received,this means that two special periodic traveling wave solutions are obtained,and then the phase diagram of the system in the local neighborhood are given.In Chapter four,the local critical periods for a Z₂-equivariant quintic system with four center conditions at the bi-center are studied.The calculation of the periodic constant of the complex system are used to prove that the order of the fine center of the system at the single equilibrium point is 4,and the determinant method is used to prove that the system has at most 4 bifurcation of critical periods from a single equilibrium point.In Chapter five,the research works of each chapter are mainly summarized and the unresolved problems in the research are also briefly described.