Estimation Of The Number Of Limit Cycles For A Class Of Planar Piecewise Linear Hamiltonian Systems Under Linear Perturbations

Determining the upper bound of the number of limit.cycles in a linear perturbation of a planar piecewise linear Hamiltonian system is one of the important research topics for weaken Hilbert’s 16th problem.The number of limit cycles of a planar piecewise linear Hamiltonian system is closely related to the number of zeros of the first order Melnikov function.The n rays:l0,l1,...,ln-1(n ≥ 2)from the origin divide the plane into n regions.Dk is the open area sandwiched between lk-1 and lk.k=1,2,...,n,ln(?)l0.Dk*(?)Dk∪lk\{(0.0)},k=2,3....,n.This paper considers the following system:(?)In which 0<ε<<1,Hk(x,y)is the two real coefficient polynomial on Dk*and Pk(x.y),Qk(x.y)are two real coefficient polynomials on the Dk*.In the first chapter,this paper introduces the research background of planar piecewise linear systems,and introduces the main conclusions:the above system can have at least[3n-3/2]limit.cycles.In the second chapter,the first order Melnikov function formula of a planar piecewise smooth Hamiltonian system under perturbation is given.The necessary assumptions.propositions and de-ductions are also given.The third chapter,the fourth chapter and the fith chapter separately calculate the first order Melnikov function as n = 2.n = 2m-1(m≥2)and n = 2m(m ≥ 2).By the Taylor expansion and the second chapter,this papaer proves that the number of the first order Melnikov function of the system can have at least[3n+3/2]zeros.and then proves the main conclusion.