The Orbit Space Approach for Piecewise Smooth Vector Fields

TL;DR

This paper develops a rigorous orbit space theory for piecewise smooth vector fields, using inverse limit techniques, and applies it to demonstrate transitivity in well-known models.

Contribution

It introduces a new, well-defined orbit space framework for PSVFs and applies it to analyze transitivity, extending previous informal approaches.

Findings

Proves the bean model is transitive in the orbit space.

Shows the sphere model is transitive in the orbit space.

Establishes a foundation for analyzing dynamical properties of PSVFs via orbit spaces.

Abstract

In this work we develop a well-defined theory of orbit spaces for piecewise smooth vector fields (PSVFs). This approach is inspired by the techniques already used in the study of endomorphisms, namely inverse limit analysis, and has been used before for PSVFs. We then apply the construction of our theory to understanding transitivity in PSVFs. Our results prove that the known examples of transitive PSVFs in the literature, the bean model and the sphere model, are indeed transitive in the orbit space.