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.