Estimation Of The Supremum Of The Number Of Limit Cycles For Two Classes Of Continuous Piecewise Linear Hamiltonian Systems Under Linear Perturbations

To determine the upper bound of the number of limit cycles of a continuous piecewise lin-ear Hamiltonian system under linear perturbations is one of the important extended topics for weakening the 16th problem of Hilbert.The number of limit cycles of continuous piecewise linear Hamiltonian system under linear disturbance is closely related to the number of isolated zeros of its first order Melnikov function.In this paper,the n parallel lines l1,l2,...,ln which parallel to Y-axis divide the plane into n+1 belt regions,which are defined as D1,D2,...,Dn+1 from left to right,where n?1.This paper considers the following system:In which 0<?<<1,Hk(x,y)is the two real coefficient polynomial on Dk*and Pk(x,y),Qk(x,y)are real coefficient polynomials of the first degree on the Dk(k=1,2,...,n+1).And on line lk,(?)Hk(x,y)/(?)x?(?)Hk+1(x,y)/(?)x,(?)Hk(x,y)/(?)y?(?)Hk+1(x,y)/(?)y(k=1,2....,n)is satisfied.In the first chapter,this paper introduces the research background of continuous piecewise linear system,and introduces two main conclusions:when n=2,if x=-1 is defined as line l1,x=1 is defined as line l2,H1(x,y)=-(x2+y2),H2(x,y)=-(y2-2x),H3(x,y)=-[(x-2)2+y2],then the upper bound of the number of limit cycles of the system across three regions under continuous linear disturbance is 2,and the upper bound of the number of limit cycles across three regions under discontinuous linear disturbance is 4;when n=3,if x=-1 is defined as line l1,x=0 is defined as line l2,x=1 is defined as line l3,H1(x,y)=-(x2+y2)=-(1+u2),H2(x,y)=-(y2-2x),H3(x,y)=-[y2-4(x+1/4)2],H4(x,y)=-[(x-6)2+y2],when u E(0,100],the upper bound of the number of limit cycles of the system across four regions under continuous linear disturbance is 5,furthermore,the upper bound of the number of limit cycles across four regions under continuous linear disturbance is greater than or equal to 5.When u?(0,30],the upper bound of the number of limit cycles of the system across four regions under discontinuous linear disturbance is 6 or 7,furthermore,the upper bound of the number of limit cycles across four regions under discontinuous linear disturbance is greater than or equal to 6.In the second chapter,the first-order Melnikov function of continuous piecewise smooth Hamil-tonian system under disturbance is given,and the necessary propositions are also given.The third chapter and the fourth chapter,the first order Melnikov function of the above sys-tem is calculated when n=2 and n=3 respectively.Through the properties of Chebyshev system and propositions in the second chapter,the two conclusions of this paper are proved respectively.