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.
Sieber, J., Kowalczyk, P., Hogan, S. J., & Di Bernardo, M. (2010). Dynamics of symmetric dynamical systems with delayed switching. Journal of Vibration and Control, 16(7-8), 1111-1140. https://doi.org/10.1177/1077546309341124