Limit Cycle Bifurcations By Perturbing A Compound Loop With A Cusp And A Nilpotent Saddle

It is well known that, The second part of 16-th problem asked by Hilbert is a most famous and greatest challenging problem of planar nonlinear di?erential equations. And the main target of a polynomial di?erential system of degree n is the number and location of limit cycles. These problems has attracted many mathematicians in China and abroad, and lots of excellent results have been obtained. Where many studies have concentrated on limit cycle bifurcations of Hamiltonian system under perturbations. For this kind of this systems,an important tool used to study Hopf bifurcation, homoclinic and heteroclinic bifurcations,Poincar′ebifurcation and nilpotent critical point bifurcation under perturbtions, e.t., is the so-called Melnikov functions or Abelian integral. In this thesis, using Melnikov function,we will study the limit cycle bifurcations of a compound loop under perturbations.In Chapter one, we introduce the background and main topics of our research. Also,we will describe the methods and main results obtained in this thesis.In Chapter two, we will study the limit cycle bifurcations of one class of nearHamiltonian system whose unperturbed system has a compound loop with a cusp, a nilpotent saddle, a homoclinic loop and two heteroclinic orbits. Using the known results and analytical technique, we obtain three expansions of Melnikov functions near the compound loop and the coe?cients in the expansions. Moreover, using these coe?cients, we may obtain a lower bound of the maximal number and the distribution of limit cycles of the system near the compound loop.In Chapter three, as an application example, we will study the number of the limit cycles for one class Li′enard system with parameters near the compound loop. First, we will give the coe?cients in the expanding expressions of Melnikov functions. Then, the limit cycles of the system near the compound loop will be studied. In precisely, we prove that the Li′enard system has at least 11, 13, 15, 18, 19 limit cycles near the compound loop respectively, when the degree of the system varies.