Limit Cycles And Bifurcations Of Local Critical Periods For Several Classes Of Differential Systems

In this paper, the problems of the limit cycles, the bifurcations of isolated periodicwave trains and the local bifurcation of critical periods for several classes of polynomialdifferential systems are studied. The thesis composed of six chapters.In Chapter one, we introduce the historical background and the present progress ofproblems for the planar polynomial differential systems. And the main work of eachchapters are concluded as well.In Chapter two, the bifurcation of limit cycles for a seven-degree polynomialdifferential system in which the origin is a3th-order nilpotent critical point is studied.The sufficient and essential conditions for the origin to be the14th-order fine focus and acenter are derived. Finally, we prove that the system has14limit cycles bifurcated fromthe origin under a small perturbation.In Chapter three, the weak centers and the local bifurcation of critical periods for acubic system are studied. The sufficient and necessary conditions of weak center areobtain. The conclusion is proved that the real system has exact3local bifurcation ofcritical periods from the origin which is satisfy with the condition of3th-order weakcenter.In Chapter four, the bifurcation of isolated periodic wave train and the monotonicityof the periodic wave solutions for a reaction-diffusion equation are studied. The equationcan be converted into a planar differential system by the wave train conversion. Usingthe algorithm, the singular point quantities and the complex period constants for theplanar differential system are calculated respectively. We get that the equation has exact5bifurcation of small amplitude isolated periodic wave trains and its periodic functionhas at most2critical periods.In Chapter five, the weak centers and the local bifurcation of critical periods for aclass of2d+3degree quasi-cubic differential system are studied. The system is transferedinto the complex system by the specific conversion. The complex period constants forthe complex system are calculated. The conditons of the weak center and isochronouscenter for the quasi differential system are obtained. Finally, we prove that the systemhas at most3bifurcation of critical periods from the origin.In Chapter six, we sum up the whole thesis and put up the prospect of the researchin the future.