Estimation Of The Supremum Of The Number Of Limit Cycles Bifurcating From A Class Of Z₂-equivariant Hamiltonian Systems Under Piecewise-smooth-polynomials Perturbations

This paper studied a class of Z2-equivariant Hamiltonian Systems (?) and when the system was perturbed by piecewise n-degree polynomials with a straight line of x=0,the paper estimated the number of its limit cycles bifurcating from the domain(?)In chapter 1,the historical background of the near-Hamiltonian vector fields and some research results of the number of limit cycles for 4th-order elliptical Hamiltonian vector fields perturbed by polynomials are introduced,as well as the common methods for studying piecewise smooth differential systems,and the main work of this paper.The major research results of this paper are as follows:(1)When n=1,2,3,for h>0,the supremum B(n)(B_c(n))of the number of limit cycles,bifuracting from the periodic closed-orbit families on the outside of the double-homoclinic orbit,for the system under the perturbation of a piecewise discontinuous(continuous)n-degree polynomial,with its first-order Melnikov function M₁(h)(M_1c(h))not equivalent to zero,is satisfied:B(1)=2,B_c(1)=0;B(2)? {3,4},B_c(2)=2;B(3)? {5,6,7,8},B_c(3)? {3,4,5,6,7}.(2)When n? 4,for h>0,the supremum B(n)(B_c(n))of the number of limit cycles,bifuracting from the periodic closed-orbit families on the outside of the double-homoclinic orbit,for the system under the perturbation of a piecewise discontinuous(continuous)n-degree polynomial,with its first-order Melnikov function M₁(h)(M_1c(h))not equivalent to zero,is satisfied:2n-1 ?B(n)?2n+7[n/2]+1,2n-3 ?B_c(n)?2n+7[n/2]-2.In Chapter 2,lemmas about constructing the first-order Melnikov function of the perturbed system are proved,and some related propositions for estimating the number of zeros of the function are mentioned.In Chapter 3,the research result(1)is proved.In Chapter 4,the research result(2)is proved.