Bifurcations Of Limit Cycles For Smooth And Nonsmooth Systems

In this thesis,we investigate bifurcations of limit cycles for smooth and nonsmooth systems.For smooth systems,we mainly study bifurcations of limit cycles for two classes of planar cubic near-Hamiltonian systems,and obtain a new lower bound for the number of limit cycles near a heteroclinic loop and a double homoclinic loop,respectively.For nonsmooth systems,we mainly study bifurcations of limit cycles for three piecewise smooth systems consisting of integrable systems with cubic reversible isochronous centers.The thesis consists of four chapters,with the specific arrangement as follows:In chapter one,we introduce the research progress and latest results of the Hilbert 16 th problem and the weakened Hilbert 16 th problem.Then,the commonly used method for studying the weakened Hilbert 16 th problem,the Melnikov function method,is introduced in detail with its higher-order form.Finally,the work and innovations of this thesis are presented.In chapter two,we mainly study the heteroclinic bifurcation of planar cubic nearHamiltonian systems through the higher-order Melnikov function.With arbitrary cubic polynomial perturbations,we calculate the asymptotic expansions of the Melnikov function near the heteroclinic loop and prove that the system can have five limit cycles around the heteroclinic loop under proper perturbations.In chapter three,we study the number of limit cycles of planar cubic near-Hamiltonian systems with a double homoclinic loop.We study a generalized Abel integral with an inverse tangent function and obtain the asymptotic expansions of the higher-order Melnikov function near the double homoclinic loop.By using the corresponding asymptotic expansion,we prove that five limit cycles can be obtained form near and outside of the double homoclinic loop.In chapter four,we mainly study three piecewise smooth systems consisting of integrable systems with cubic reversible isochronous centers.Under arbitrarily switch lines passing through the origin,we give the center conditions of the three systems by computing the Lyapunov constants.For three piecewise smooth systems,we prove that the maximal number of small-amplitude limit cycles are 3,4 and 4,respectively.