The Melnikov method and subharmonic orbits in a piecewise smooth system

TL;DR

This paper analyzes the existence of subharmonic orbits in a two-dimensional piecewise smooth Hamiltonian system with impacts, using Melnikov methods to establish conditions for periodic orbits under perturbations.

Contribution

It extends Melnikov theory to piecewise smooth Hamiltonian systems with impacts, providing rigorous conditions for subharmonic orbit existence and heteroclinic splitting.

Findings

Existence of $nT$-periodic impacting orbits under small periodic perturbations.

Conditions for the existence of impacting orbits when crossing discontinuities are large.

Criteria for heteroclinic connection splitting in the system.

Abstract

In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold $x=0$. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the system. We also suppose that the system possesses an invisible fold-fold at the origin and two heteroclinic orbits connecting two hyperbolic critical points on either side of $x=0$. Finally, we assume that the region closed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they approach the heteroclinic connection. When considering a non-autonomous ($T$-periodic) Hamiltonian perturbation of amplitude $\varepsilon$, using an impact map, we rigorously prove that, for every $n$ and $m$ relatively prime and $\varepsilon>0$ small enough, there exists a $nT$-periodic orbit impacting $2m$ times with the switching manifold at every period if a modified subharmonic Melnikov function possesses a simple zero. We also prove that, if the orbits are discontinuous when they cross $x=0$, then all these orbits exist if the relative size of $\varepsilon>0$ with respect to the magnitude of this jump is large enough. We also obtain similar conditions for the splitting of the heteroclinic connections.