Chaos explosion and topological horseshoe in three-dimensional impacting hybrid systems with a single impact surface

Abstract

For a class of three-dimensional impacting hybrid systems comprising a linear system of ordinary differential equations and a reset map, and having a single impact surface, this paper studies the phenomena of impacting homoclinic bifurcation leading to periodic orbits, horseshoes and chaos explosions. More precisely, it is proved that the homoclinic bifurcation can result in the impacting periodic sinks or impacting periodic saddle orbits when the impact surface is a plane and the reset map satisfies some basic conditions, but it is not easy to find horseshoes in this case. Furthermore, when the single impact surface is not a plane and the reset map has a certain rotational property, it is proved that a topological horseshoe will appear suddenly when bifurcation parameter passes through some threshold, thus, a kind of chaos explosion takes place. These main results are illustrated by several examples, in which it can be seen that the newborn chaotic invariant sets might be chaotic attractors from the perspective of numerical simulation.