Up to second order Melnikov functions for general piecewise Hamiltonian systems with nonregular separation line

Abstract

In this paper, we consider a perturbation of planar general piecewise Hamiltonian systems with nonregular separation line. A general explicit expression of the first and second order Melnikov functions is presented for such piecewise Hamiltonian systems. In addition, the maximum number of limit cycles for piecewise linear differential system is an important topic that many researchers are concerned with. Different from considering a piecewise linear perturbation of a linear center in the previous research, we apply our explicit expression to estimate the number of limit cycles bifurcated from a class of piecewise linear Hamiltonian systems. More specifically, the unperturbed system we study is piecewise smooth and constructed by a linear center and a constant differential system, which is difficult to be considered by a polar coordinate transformation. And we obtain the upper bounds to be 4 and 5 for the limit cycles bifurcated from the periodic orbits by using the first and second order Melnikov functions, respectively. Moreover the upper bounds are sharp.