Existence of Periodic Orbits with Zeno Behavior in Completed Lagrangian Hybrid Systems

Abstract

In this paper, we consider hybrid models of mechanical systems undergoing impacts, Lagrangian hybrid systems, and study their periodic orbits in the presence of Zeno behavior-an infinite number of impacts occurring in finite time. The main result of this paper is explicit conditions under which the existence of stable periodic orbits for a Lagrangian hybrid system with perfectly plastic impacts implies the existence of periodic orbits in the same system with non-plastic impacts. Such periodic orbits contain phases of constrained and unconstrained motion, and the transition between them necessarily involves Zeno behavior. The result is practically useful for a wide range of unilaterally constrained mechanical systems under cyclic motion, as demonstrated through the example of a double pendulum with a mechanical stop.