Symbolic dynamics of planar piecewise smooth vector fields

TL;DR

This paper explores the chaotic behavior of planar piecewise smooth vector fields (PSVFs), demonstrating their unique properties compared to smooth vector fields and establishing a conjugation with shift maps to analyze their dynamics.

Contribution

It introduces a novel conjugation between PSVFs and shift maps, providing a new perspective for analyzing chaos in PSVFs, which differ fundamentally from smooth vector fields.

Findings

Chaotic behavior is possible in planar PSVFs, unlike smooth vector fields.

A conjugation between PSVFs and shift maps is constructed.

New analytical methods for PSVFs are developed based on discrete dynamics.

Abstract

Recently, the theory concerning piecewise smooth vector fields (PSVFs for short) have been undergoing important improvements. In fact, many results obtained do not have an analogous for smooth vector fields. For example, the chaoticity of planar PSVFs, which is impossible for the smooth ones. These differences are generated by the non-uniqueness of trajectory passing through a point. Inspired by the classical fact that one-dimensional discrete dynamic systems can produce chaotic behavior, we construct a conjugation between the shift map and PSVFs. By means of the results obtained and the techniques employed, a new perspective on the study of PSVFs is brought to light and, through already established results for discrete dynamic systems, we will be able to obtain results regarding PSVFs.