Limit sets of discontinuous vector fields on two-dimensional manifolds

TL;DR

This paper analyzes the long-term behavior of trajectories in discontinuous vector fields on two-dimensional manifolds, classifying their limit sets and exploring phenomena like sliding motion and chaos.

Contribution

It provides a comprehensive classification of limit sets and analyzes the dynamics of discontinuous vector fields on 2D manifolds, including sliding motion and non-recurrent behavior.

Findings

Classification of possible limit sets

Existence of sliding motion on manifolds

Presence of nondeterministic chaos

Abstract

In this paper the asymptotic behavior of trajectories of discontinuous vector fields is studied. The vector fields are defined on a two-dimensional Riemannian manifold $M$ and the confinement of trajectories on some suitable compact set $K$ of $M$ is assumed. The behavior of the global trajectories is fully analyzed and their limit sets are classified. The presence of limit sets having non-empty interior is observed. Moreover, the existence of the so called sliding motion is allowed on $M$. The results contemplate a list of possible limit sets as well the existence of non-recurrent dynamics and the presence of nondeterministic chaos. Some examples and classes of systems fitting the hypotheses of the main theorems are also provided in the paper.