2008/04/02 by J. Sieber, P. Kowalczyk, S. J. Hogan +3 · 18 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Chaos control and synchronization #Class (philosophy) #Dynamical systems theory #Focus (optics) #Invariant (physics) #Piecewise #Piecewise linear function #Quantum chaos and dynamical systems #Reflection (computer programming) #Reflection symmetry #Torus #math.DS #math.OC #msc:37G15 #msc:70K43 #msc:70K50
paper · pdf · doi:10.1177/1077546309341124
published in Journal of Vibration and Control 16(7-8), 1111-1140 (SAGE Publishing) · 28 pages
arxiv created 2008/04/02 · openalex publication_date 2010/05/19 · arxiv updated 2010/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.