Higher order Melnikov functions for piecewise smooth differential systems with four zones

Abstract

In this work, we develop a systematic algorithm for computing Melnikov functions of arbitrary order for planar piecewise smooth Hamiltonian systems separated by the coordinate axes and perturbed within the class of polynomial differential systems. Two equivalent formulations of the Melnikov functions are obtained. The first formulation involves the flight times and divergence integrals of certain vector fields along the trajectories of the underlying Hamiltonian system, while the second formulation avoids the explicit use of flight times and trajectory expressions. By applying the resulting Melnikov functions, we determine the number of limit cycles bifurcating from planar piecewise Hamiltonian systems with four regions under polynomial perturbations.