Homoclinic Theorems for piecewise smooth vector fields

Claudio Buzzi; Paulo Santana; Luan V. M. F. Silva

Homoclinic Theorems for piecewise smooth vector fields

Dynamical Systems 2026-07-10 v1

Authors: Claudio Buzzi , Paulo Santana , Luan V. M. F. Silva

Abstract

In this paper we provide extensions of the Birkhoff-Smale Homoclinic Theorem and the $\lambda$ -Lemma for piecewise smooth vector fields free of sliding motion. We prove that if the vector field is $T$ -periodic in its independent variable and has a transverse homoclinic point, then its time- $T$ -map is conjugated with a Bernoulli Shift and thus the vector field is chaotic. The Bernoulli Shifts considered here may have any finite number of symbols, or countably infinite many.

Cite

@article{arxiv.2607.09618,
  title  = {Homoclinic Theorems for piecewise smooth vector fields},
  author = {Claudio Buzzi and Paulo Santana and Luan V. M. F. Silva},
  journal= {arXiv preprint arXiv:2607.09618},
  year   = {2026}
}