Limit Cycle Bifurcations Near A Double Homoclinic Loop Or A Heteroclinic Loop With Nilpotent Saddles

Differential equation theory has important applications in medicine,economics,biology,and other disciplines.The qualitative theory of differential equations is one of the important research fields of differential equations.In this paper,we mainly discuss two kinds of double homoclinic loop and heteroclinic loop bifurcations with nilpotent saddles.In Chapter 1,we outline the research background,current situation,and main research content of this article.In Chapter 2,we discuss the conditions for the generation of limit cycles in a center-symmetrical near-Hamiltonian system near a double homoclinic loop of cuspidal type with one nilpotent saddle of order two,and apply the obtained results to a kind of centrally symmetric Liénard system to study the problem of the number of limit cycles,finally,we give a lower bound for the maximum number of limit cycles in the system.In Chapter 3,we study the calculation formulas of coefficients in the first order Melnikov function expansion near a heteroclinic loop with two arbitrary order nilpotent saddles for a near-Hamiltonian system under certain conditions.We further give the general conditions for obtaining limit cycles near the heteroclinic loop,and apply the obtained results to a specific system to study the number of limit cycles.Using the results obtained in this paper,we can study the number of limit cycles near the heteroclinic loop of the Hamiltonian system containing a heteroclinic loop with nilpotent saddles under higher order perturbations.