Dynamics of symmetric dynamical systems with delayed switching

J Sieber, PS Kowalczyk, SJ Hogan, M di Bernardo

Research output: Working paperWorking paper and Preprints

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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 Poincar 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.
Original languageEnglish
Publication statusAccepted/In press - 12 Dec 2007

Bibliographical note

Sponsorship: The research of J.S. and P.K. was partially supported by by EPSRC grant GR/R72020/01.

Keywords

  • hysteresis
  • relay
  • invariant torus collision
  • delay

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