Bifurcation Of Limit Cycles From Heteroclinic Loops With A Cusp In Some Planar Polynomial Systems

As an introduction, in the first chapter we introduce the background of our research andmain topics that we will study in the following chapters. We also give a description of ourmethods and results detained in this thesis in the first chapter.In the second chapter, we study the expansion of Melnikov function near a heteroclinicloop with more than two cusps for some near Hamiltonian system, we give the formula for thefirst coeficients of the expansion, we also give the method to find the limit cycles using thesecoeficients. As we know, there are some results about the expansion of Melnikov functionnear the homoclinic loop with a saddle or a cusp and the heteroclinic loop with more thantwo saddles, the formula for the first coeficients have been given. In this chapter, we firstlystudy the expansion of Melnikov function near a heteroclinic loop with two cusps, then forthe case of more than two cusps, the method for the bifurcation of limit cycles is also given.In the third chapter, we study the expansion of Melnikov function near a heteroclinicloop with a hyperbolic saddle and a cusp for some near-Hamiltonian system, we obtain anexplicit formula to compute the first coeficients, the method to find the limit cycles usingthese coeficients. We also present some interesting applications.In the fourth chapter, using Han Maoan's method, we study the number of limit cyclesof Z4 quintic near-Hamiltonian system, we obtain that there can be 28 limit cycles bifurcatedfrom the homoclinic loop and the center. At last, using the hopf bifurcation theory, we find40 small limit cycles from a Z4 seventh order near-Hamiltonian system.