Dynamics of symmetric dynamical systems with delayed switching

Sieber, J, Kowalczyk, P, Hogan, SJ and Di Bernardo, M (2010) Dynamics of symmetric dynamical systems with delayed switching. Journal of Vibration and Control, 16 (7-8). pp. 1111-1140. ISSN 1077-5463

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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 Poincaré map near the colliding periodic orbit. The Poincaré 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. © 2010 SAGE Publications.