Limit Cycles Near Homoclinic And Double Homoclinic Loops

As an introduction, in the first chapter we introduce the background of our research and main topics that we will study in the following chapters. We also give a description of our methods and results detained in this thesis in the first chapter.In the second chapter, our main purpose is to give an explicit formula to compute the first four coefficients appeared in the expansion of the first order Melnikov function at the Hamiltonian value h₀ such that the curve defined by the equation H(x, y) = h₀ contains a homoclinic loop, where the formula for the fourth coefficient is new, and to give a way to find limit cycles near the loops by using these coefficients.In the third chapter, we study the expansion of the first order Melnikov function of a near-Hamiltonian system on the plane near a double homoclinic loop, obtain an explicit formula to compute the first four coefficients, and then give a way to find at least 7 limit cycles near the double homoclinic loop by using these coefficients. We also present some interesting applications.In the fourth chapter, we study the number of a kind of third order polynomial system near a homoclinic loop using the method in chapter 2. We can get 5 limit cycles near the homoclinic loop, improving some known results.