Invariant Forms in Hybrid and Impact Systems and a Taming of Zeno

TL;DR

This paper establishes conditions for invariant forms in hybrid and impact systems, showing that invariant volumes can prevent Zeno behavior and ensure recurrence in systems like billiards with continuous and discrete dynamics.

Contribution

It provides necessary and sufficient conditions for invariance of differential forms, especially invariant volumes, in hybrid systems with impact dynamics, linking invariance to Zeno behavior suppression.

Findings

Invariant volume forms inhibit Zeno trajectories in hybrid systems.

Many billiard systems are recurrent regardless of table shape.

Hybrid impact systems with boundary identity property tend to avoid Zeno phenomena.

Abstract

Hybrid (and impact) systems are dynamical systems experiencing both continuous and discrete transitions. In this work, we derive necessary and sufficient conditions for when a given differential form is invariant, with special attention paid to the case of the existence of invariant volumes. Particular attention is given to impact systems where the continuous dynamics are Lagrangian and subject to nonholonomic constraints. A celebrated result for volume-preserving dynamical systems is Poincar\'e recurrence. In order to be recurrent, trajectories need to exist for long periods of time, which can be controlled in continuous-time systems through e.g. compactness. For hybrid systems, an additional mechanism can occur which breaks long-time existence: Zeno (infinitely many discrete transitions in a finite amount of time). We demonstrate that the existence of a smooth invariant volume severely inhibits Zeno behavior; hybrid systems with the "boundary identity property" along with an invariant volume-form have almost no Zeno trajectories (although Zeno trajectories can still exist). This leads to the result that many billiards (e.g. the classical point, the rolling disk, and the rolling ball) are recurrent independent on the shape of the compact table-top.