Estimation Of The Number Of Limit Cycles On A Class Of Hamiltonian Systems With A Global Center Under Piecewise Polynomial Perturbation

This paper studied a class of Hamiltonian systems with a global centerand estimated the number of its limit cycles when it was perturbed by piecewise n-degree polyno-mials with a straight line of y=0.In chapter 1,the historical background of the limit cycles are introduced.Some research results of the number of limit cycles for some elliptical and hyper-elliptical Hamiltonian vector fields under polynomial perturbation are summarized,and also introducing the main work of this paper.The major research results of this paper are as follows:The lowest upper bounds for the number of limit cycles of the system under the perturbation of discontinuous piecewise polynomials of degree n（n（?）N₊）are B（n）and continuous piecewise polynomials of degree i（i=1,2,3）are B_c（i）（i=1,2,3）.Satisfing（1）B（1）=1,3≤B（2）≤4,5≤B（3）≤9,When n≥4,n+2≤B（n）≤6n 2[1+（1）ⁿ].（2）B_c（1）=0,B_c（2）=1,3≤B_c（3）≤5.In chapter 2,the lemmas about constructing the first-order Melnikov function of the perturbed system are proved.And other lemmas related to estimating the number of the zero point of the Melnikov function are introduced,such as second-order differential operators and Descartes sign rule etc.In chapter 3,the first-order Melnikov function M₁（h）of the system under the perturbation of discontinuous piecewise polynomials of degree n is calculated,and the independence of the coe cient polynomial of the generator with respect to the perturbation polynomial is demonstrated.In chapter 4,the research result（1）is proved.In chapter 5,the research result（2）is proved.