Dynamics of symmetric dynamical systems with delayed switching

Details Dynamics of symmetric dynamical systems with delayed switching Dynamics of symmetric dynamical systems with delayed switching

2008-04-02

arxiv.org/abs/0804.0408v1

Journal of Vibration and Control 16(7-8) pp. 1111-1140, 2010

Mathematics

Dynamical Systems

28 pages

Scientific paper

10.1177/1077546309341124

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.

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