Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation

TL;DR

This paper analyzes bifurcations of limit cycles in a piecewise smooth Hamiltonian system with boundary perturbations, deriving Melnikov functions to determine the number of bifurcating limit cycles.

Contribution

It provides explicit formulas for first and second order Melnikov functions in a general second order perturbation of such systems, linking limit cycle count to zeros of a boundary function.

Findings

Number of limit cycles equals the number of positive zeros of boundary function f at zero perturbation.

Derived explicit expressions for Melnikov functions for the system.

Established a direct relation between boundary perturbation zeros and bifurcating limit cycles.

Abstract

In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general second order perturbation, which can be used to find the number of limit cycles bifurcated from periodic orbits. Further, we have shown that the number of limit cycles of the system $\dot{X}=\begin{cases} (H_y^+,-H_x^+) & \mbox{if}~y>\varepsilon f(x)\\ (H_y^-,-H_x^-) & \mbox{if}~y<\varepsilon f(x) \end{cases}$ equals to the number of positive zeros of $f$ when at $\varepsilon=0$ the system has a period annulus around the origin.