The Number Of Limit Cycles For Segmented Secondly Near Hamiltonian Systems

As we all know, the bifurcation theory of limit cycles of planar systems is the main topic of di?erential equation theorem and the most famous problem is the D.Hilbert’s 16 th problem. These questions have been drawing many excellent mathematicians’ attention who gain many investigative issues. Recently, influenced by the actual problem, piecewise smooth systems have been paid attention by many mathematicians, and scientists gained many research results.In this paper, we mainly investigate the number of limit cycles of a class of piecewise quadratic near-Hamiltonian system.In Chapter one, we introduce the background of our research and the main topics that we will study in the following chapters. In Chapter two, we discuss some basic theory that relate to this paper and the theorems or lemmas applied in the paper. In Chapter three, we are concerned with the number of maximal number of limit cycles for a quadratic near-Hamiltonian system by a new method of bifurcation theory. First of all, we give the possible portraits with two saddles and a generalized closed orbit when the system isn’t perturbed. Secondly, we study the number of limit cycle of perturbed system when the corresponding unperturbed system with two saddles. In other word, by using the method of generalized Melnikov functions to study the expansion near the generalized cycles we gain the number of limit cycles.