On the Existence and Uniqueness of Poincar\'e Maps for Systems with Impulse Effects

TL;DR

This paper establishes conditions ensuring the existence and uniqueness of Poincaré maps in impulsive dynamical systems on manifolds, extending their application to systems with multiple domains.

Contribution

It provides new theoretical conditions for the existence and uniqueness of Poincaré maps in systems with impulse effects on manifolds, including multi-domain cases.

Findings

Derived sufficient conditions for Poincaré map existence.

Proved uniqueness of Poincaré maps under these conditions.

Applied results to systems with multiple domains.

Abstract

The Poincar\'e map is widely used to study the qualitative behavior of dynamical systems. For instance, it can be used to describe the existence of periodic solutions. The Poincar\'e map for dynamical systems with impulse effects was introduced in the last decade and mainly employed to study the existence of limit cycles (periodic gaits) for the locomotion of bipedal robots. We investigate sufficient conditions for the existence and uniqueness of Poincar\'e maps for dynamical systems with impulse effects evolving on a differentiable manifold. We apply the results to show the existence and uniqueness of Poincar\'e maps for systems with multiple domains.