Estimation Of The Lower Bound Of Limit Cycles For Four Classes Of Planar Piecewise Linear Hamiltonian Systems Under Polynomial Perturbation

The question of the ruunber of limit cycles was raised by Ponicare.In 1900,the latter part of the 16th problem was raised by mathematician Hilbert at the annual meeting of International Mathematics:Seting Fn(x,y),Gn(x,y)be a real coefficient polynomial which with the degree less than n,the maximum number N(n)of Limit Cycles of the planar polynomial system with the degree of n and the relative position.In this paper,the author studies the lower bound of the number of limit cycles under the polynomial perturbation of a planar piecewise linear Hamiltonian system that divides the plane into ?(?=2.3,4,5)sectoral regions.By using first-order Melnikov function and Taylor expansion method,the lower bound of the limit cycles number of piecewise linear Hamiltonian system divided into different sector regions under polynomial disturbance is obtained.The ? rays:l0,l1,...,l?-1 from the origin divide the plane into n sectors.Dk is the open sandwiched between l?-1 and l?,?=1,2,...,k,ln(?)l0.the following system is studied in this paper.In the first chapter,the author introduces the research background of planar piecewise linear systems and introduces the main conclusions:the lower bound of limit cycles for four kinds of plane piecewise linear Hamiltonian systems with polynomial perturbations of degree n is studied in this paper.In the second chapter,the corresponding lemmas,propositions and corollaries used in theorem proof are given.In the third chapter,the fourth chapter,the fifth chapter and the sixth chapter,the author respectively calculate the above systems when ?=2,?=3,?=4 and ?=5.The first-order Melnikov function of the system is estimated by Taylor expansion and the inference in Chapter 2.The lower bound of the number of limit cycles for piecewise linear Hamiltonian system under the perturbation of n-th polynomial is given.