Shilnikov problem in Filippov dynamical systems

TL;DR

This paper introduces the concept of sliding Shilnikov orbits in 3D Filippov systems, proving their generic occurrence and connection to infinite sliding periodic orbits, with the first examples in discontinuous linear systems.

Contribution

It defines sliding Shilnikov orbits in Filippov systems, proves a Shilnikov-type theorem, and provides the first examples in discontinuous linear systems.

Findings

Sliding Shilnikov orbits occur in generic one-parameter families.

Near a sliding Shilnikov orbit, there are countably infinite sliding periodic orbits.

First known examples of such phenomena in Filippov systems.

Abstract

In this paper we introduce the concept of sliding Shilnikov orbits for $3$D Filippov systems. In short, such an orbit is a piecewise smooth closed curve, composed by Filippov trajectories, which slides on the switching surface and connects a Filippov equilibrium to itself, namely a pseudo saddle-focus. A version of the Shilnikov's Theorem is provided for such systems. Particularly, we show that sliding Shilnikov orbits occur in generic one-parameter families of Filippov systems, and that arbitrarily close to a sliding Shilnikov orbit there exist countably infinitely many sliding periodic orbits. Here, no additional Shilnikov-like assumption is needed in order to get this last result. In addition, we show the existence of sliding Shilnikov orbits in discontinuous piecewise linear differential systems. As far as we know, the examples of Fillippov systems provided in this paper are the first exhibiting such a sliding phenomenon.