Existence of Periodic Orbits in Completed Lagrangian Hybrid Systems with Non-Plastic Collisions

In this paper, we consider hybrid models of mechanical systems undergoing impacts, i.e., Lagrangian hybrid systems, and study their periodic orbits in the presence of Zeno behavior. The main result of this paper is explicit conditions under which the existence of stable periodic orbits for a Lagrangian hybrid system with plastic impacts implies the existence of periodic orbits in the same Lagrangian hybrid systems with non-plastic impacts. Since non-plastic impacts result in Zeno behavior, in proving this result we necessarily obtain an understanding of periodic orbits containing Zeno behavior. These results are practically useful to a wide range of mechanical systems, as demonstrated through the example of a double pendulum with a mechanical stop.