Homoclinic Boundary-Saddle Bifurcations in Planar Nonsmooth Vector Fields

Abstract

In a smooth dynamical system, a homoclinic connection is an orbit connecting a saddle equilibrium to itself. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interactions 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 to cross or slide along the discontinuity. Even the simplest case of connection to a regular saddle, which hits a discontinuity as a parameter is varied, is surprisingly complex. In this paper, we construct bifurcation diagrams for nonresonant saddles in the plane, unfolding the homoclinic connection to a boundary saddle in a nonsmooth dynamical system. As an application, we exhibit such diagrams for a model of a forced pendulum.