The Studies Of Bifurcation Of Limit Cycles For Several Classes Of Polynomial Differential Autonomous Systems

The problems about bifurcations of limit cycles for polynomial differential autonomy systems are investigated in this paper. It is composed of five chapters.In Chapter one, the historical background about bifurcations of limit cycles for planar autonomy differential systems is introduced and the main works of this paper are concluded as well.In Chapter two, there are two sections. In Section one, the bifurcations of limit cycles at the origin for a class of 2n+1-degree polynomial differential system are investigated. By computation and deducting in theories, the expressions of singular points of the system at origin are obtained. Then the conditions of center and the highest fine focal points are gotten. Finally, an example of polynomial differential system with four limit cycles bifurcating from the origin is constructed. In Section two, the bifurcations of limit cycles at the infinity for the polynomial differential system are studied. We obtain the conditions of the center and the highest-order weak focus. Then an example of polynomial system with three limit cycles at infinity is given.In Chapter three, an indirect method is used to investigate the bifurcations of limit cycles at infinity for a class of seventh-degree polynomial system. The conclusion of a seventh-degree differential system can bifurcate eleven limit cycles at infinity is proved. It's the first time we get such a good result. In Chapter four, the bifurcations of limit cycles in the region of local neighborhood for a class of fifth-degree symmetrical polynomial system are studied. By computation of singular point values, the focus values are gotten. Then we prove that the system has sixteen small amplitude limit cycles. The proof of existence of limit cycles is algebraic and symbolic. And the phase portrait in the region of local neighborhood is showed.In the last chapter, the whole paper is summed up and the problems which are still unsolved in the research are showed.