Estimation Of The Number Of Limit Cycles For Six Class Of Piecewise Linear Hamiltonian Systems Under Polynomial Perturbation

Recently,piecewise smooth dynamical systems have been widely studied and applied in the fields of mechanics,electronic engineering and automation theory.Piecewise smooth system is a type of nonlinear smooth system,in this paper,we mainly study the problem of the number of limit cycles of six class of piecewise linear Hamiltonian systems by perturbing with n polynomial.By calculating the expression of the first-order Melnikov function and using the generalized Rolle theorem,this paper discussed range of the maximal number of limit cycles six classes of piecewise smooth quadratic planar systems with parabolic-parabolic type,parabolic-ellipse type,parabolic-hyperbolic type,ellipse-ellipse type,ellipse-hyperbolic type and hyperbolic-hyperbolic type.This article is divided into four chapters and the main contents are summarized as follows:The first chapter introduced the background of the research,the progress of the research and the main results of this paper.The second chapter is the preparatory knowledge,which mainly introduced several lemmas related to this article.In the third chapter,we studied the first-order Melnikov functions of linear Hamilton systems under the perturbation of n polynomials on the half plane with parabolic type,elliptic type and hyperbolic type respectively.In the fourth chapter,we discussed the range of the maximal number Z(n)of limit cycles about six classes of piecewise linear Hamiltonian systems with parabolic-parabolic type,parabolic-ellipse type,parabolic-hyperbolic type,ellipse-ellipse type,ellipse-hyperbolic type and hyperbolic-hyperbolic type,by the expression of first-order Melnikov function and the generalized Rolle the-orem,we estimated the range of Z(n)by using the relation between the number of roots and the number of limit cycles.