Bifurcation Of Limit Cycles Near Homoclinic Loops With Degenerate Singularities

The first Chapter is an introduction,which mainly introduces the significance and pur-pose of the research topic,the current situation at home and abroad,as well as the research content and results of this paper.In Chapter 2,we mainly study the lower bound of the maximum number of limit cycles for a class of quadratic reversible Hamiltonian systems with non-Morse point under high-order small perturbations.In this Chapter,according to the structured simple perturbation terms of Hamiltonian function,the analysis of higher order generator of y in Melnikov func-tion is simplified.By asymptotic analysis of generator x~iy,its asymptotic expansion near homoclinic loop is given.Applying Tian-Han’s idea of solving the higher order terms of the expansion,the coe cients of the terms whose degree is greater than 2 in the expansion are obtained.Based on the mutual independence of the coe cients,the corresponding number of limit cycles are obtained.By making full use of the perturbation parameters,a lower bound of the number of the system’s limit cycles is obtained,which enriches the research results of the general polynomial perturbation of quadratic reversible Hamiltonian systems.In Chapter 3,we study the expansion of first order Melnikov functions near homo-clinic loops with degenerate singularities on invariant straight lines,and provide a general asymptotic expression for general Hamiltonian systems.For homoclinic loops containing non degenerate saddle points,nilpotent saddle points,and nilpotent cusps,the expression of Melnikov function expansions have been given by scholars.However,there are no relevant results for homoclinic loops connecting degenerate singularities.In this Chapter,through a series of nonlinear transformations,the system is transformed into a locally symmetric system,which provide a calculation basis for the analysis of the asymptotic expansions of general generator,and give the general forms of Melnikov function expansions near homoclin-ic loops with multiple types of degenerate singular points connected with invariant straight linesIn Chapter 4,we mainly use the theoretical results in Chapter 3 to study two kinds of Hamiltonian systems with specific perturbations,give the corresponding Melnikov function expansions,and explore the number of limit cycles of two kinds of polynomial systems by determining the zero point of the corresponding Melnikov function.In Chapter 5,this section summarizes the scientific issues studied and proposes further research questions.