A Piecewise Smooth $\lambda$-Lemma

TL;DR

This paper extends the classical $mbda$-Lemma to piecewise smooth vector fields and maps, analyzing the regularity of time-T-maps and establishing their homeomorphism and diffeomorphism properties.

Contribution

It introduces new extensions of the $mbda$-Lemma for piecewise smooth systems and studies the regularity of associated time-T-maps.

Findings

Time-T-maps are homeomorphisms.

Time-T-maps are piecewise smooth diffeomorphisms.

Extensions of the $mbda$-Lemma to piecewise smooth systems.

Abstract

In this paper we provide extensions of the $\lambda$-Lemma (also known as Inclination Lemma) for piecewise smooth vector fields and maps. In order to achieve our main result, we investigate the regularity of time-T-maps of piecewise smooth vector fields defined at crossing orbits. We prove that these maps are homeomorphisms and also piecewise smooth diffeomorphisms.