Dynamics of symmetric dynamical systems with delayed switching

TL;DR

This paper investigates how delays in switching between vector fields affect symmetric dynamical systems, classifying event collisions and analyzing their bifurcations, with applications to oscillators and invariant tori.

Contribution

It provides a new classification and analysis of event collisions in delayed switching systems with symmetry, deriving an implicit Poincare map expression.

Findings

Event collisions cause changes in the Poincare map's dimension.

Symmetric periodic orbits exhibit specific bifurcation behaviors at collisions.

Attracting invariant polygons emerge at torus collisions.

Abstract

We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric periodic orbits in systems with full reflection symmetry, a symmetry that is prevalent in applications. We derive an implicit expression for the Poincare map near the colliding periodic orbit. The Poincare map is piecewise smooth, finite-dimensional, and changes the dimension of its image at the collision. In the second part of the paper we apply this general result to the class of unstable linear single-degree-of-freedom oscillators where we detect and continue numerically collisions of invariant tori. Moreover, we observe that attracting closed invariant polygons emerge at the torus collision.