Bifurcations Of Limit Cycles For Several Classes Of Differential Systems With Parameters

In this thesis,we study the bifurcations of limit cycles for three kinds of planar differential systems,a class of piecewise smooth systems with a center and a homoclinic loop,a kind of quadratic isochronous systems with switching lines and a class of nearHamiltonian systems with two small parameters,respectively.In this paper,the Melnikov function method is mainly used to study the bifurcations of limit cycles for three classes of systems.In addition,the Picard-Fuchs method and Chebyshev criterion are also used.This paper is divided into four chapters and is organized as follows:Chapter one is an introduction,which introduces the background knowledge,development process,research methods and research status of the research subject.In chapter two,we study the bifurcations of limit cycles arising after perturbations of a special piecewise smooth system,which has a center and a homoclinic loop.By using the Abelian integrals and Melnikov functions,we obtain an upper bound of the maximum number of limit cycles bifurcated from the period annulus between the center and the homoclinic loop.Furthermore,by applying the expansion of the first order Melnikov function we give a lower bound of the maximum number of limit cycles bifurcated from the center.In chapter three,we study the maximal number of limit cycles bifurcating from the period annuli of a quadratic isochronous system with piecewise 9)-th polynomial perturbations.By using high-order derivative method,for 9)≥ 3 we obtain an upper bound of the number of limit cycles bifurcated from period annuli of system.Furthermore,for 9)= 1,2we give the least upper bound of the number of limit cycles bifurcated from period annuli of system by using Chebyshev criterion.The above results improve the conclusions in [80].In chapter four,we consider the bifurcations of limit cycles for a class of nearHamiltonian systems with multiple parameters.By expanding the first order Melnikov function for the small parameter ,we obtain that 5 limit cycles can be generated in the period annulus near the origin.