Homoclinic boundary-saddle bifurcations in nonsmooth vector fields

TL;DR

This paper investigates complex bifurcation phenomena involving homoclinic connections in nonsmooth dynamical systems, highlighting how discontinuities influence bifurcations and chaos, with detailed diagrams and an example in a forced pendulum.

Contribution

It provides the first detailed bifurcation diagrams for homoclinic connections to saddles in nonsmooth systems, including interactions with discontinuities and an example in a forced pendulum.

Findings

Bifurcation diagrams for non-resonant saddles in nonsmooth systems.

Complex interactions between homoclinic connections and discontinuities.

Illustrative example involving a forced pendulum.

Abstract

In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interaction with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection allowed to cross or slide along the discontinuity. Even the simplest case, that of connection to a regular saddle that hits a discontinuity as a parameter is varied, is surprisingly complex. Bifurcation diagrams are presented here for non-resonant saddles in the plane, including an example in a forced pendulum.