Bifurcation Of Heteroclinic Loop And Double Heteroclinic Loop With A Nilpotent Cusp And A Hyperbolic Saddle

Differential dynamical system is an important brach of mathematics.With the development of science and technology,the relevant theories of differential dynamical system have important applications in the study of trajectory of cetestial bodies,the spread of epidemic diseases,etc..In this paper we mainly study limit cycles bifurcations of two kinds of heteroclinic loop and double heteroclinic loop with a nilpotent cusp and a hyperbolic saddle.In Chapter one,we mainly introduce the research background,research status and the main methods which we used.In Chapter two,we first introduce some preknowledge.Then,we study the expansion of Melnikov function near a heteroclinic loop with a hyperbolic saddle and a nilpotent cusp of order m.In addition,we give the formula for local coe cients in this expansion.Mainly,we study the calculation formulas of all analytic coe cients under certain conditons in this expansion of Melnikov function and further give the conditions to obtain limit cycles near the heteroclinic loop by using these coe cients.Finally,we use the results obtained to study the number of limit cycles for a Liénard system of degree 6 under a polynomial perturbation.In Chapter three,we study the coe cients formulas of all analytic functions in the expansion of Melnikov function near a double heteroclinic loop consisting of two nilpotent cusps and a hyperbolic saddle,and give the conditions to obtain limit cycles near the double heteroclinic loop.Further,by using the results given in this paper,we study the number of limit cycles of a class of centrally symmetric Liénard systems.In this paper,we study more coe cients in the expansions of Melnikov functions near a heteroclinic loop and a double heteroclinic loop,so that we can study the bifurcations of near Hamiltonian systems with this kind of odd closed orbit under higher perturbations.