Hopf Bifurcation For A Planar Quadratic Integrable System Under Smooth And Piecewise Smooth Perturbations With Multiple Parameters

By using the first order Melnikov function method with multiple parameters,in this paper we study Hopf bifurcation for a planar quadratic integrable system with three cases of perturbation terms which include a quadratic polynomial,an an-alytic expression with singular terms and a piecewise smooth quadratic polynomial,obtaining 2,3 and 7 limit cycles,respectively.Under quadratic piecewise smooth polynomial perturbations of a planar quadratic integral system more limit cycles are found than the known results.Our results also show that singular perturbation-s usually play important role in limit cycle bifurcations.This paper includes the following four chapters:In the first chapter we introduce Hilbert 's 16th problem and its weak form,and the main idea and innovation of the paper.In the second chapter we recall the first order Melnikov methods for planar smooth and piecewise smooth near Hamiltonian systems with multiple parameters and some relevant formulas.In the third chapter,when the perturbation terms are a quadratic polynomial and an analytic expression with singular terms,by multiplying an integral factor the near integrable system considered is transformed into a near Hamiltonian system.Then the expressions of the first order Melnikov function in two cases are given.The analytic properties and the estimation of the number of zero with the Melnikov function in the corresponding interval are studied,and the lower bounds of the number of limit cycles in the Hopf bifurcation of the system are obtained.In the fourth chapter,when the perturbation term is a piecewise smooth polyno-mial with degree two,we apply the first order Melnikov function method of piecewise near-Hamiltonian systems with multiple parameters to analyze the analytic proper-ties and the number of zeros of the first order Melnikov function,obtaining a new result that a quadratic piecewise polynomial system with two zeros separated by a straight line can has 7 limit cycles in Hopf bifurcation.