On the non-existence of isochronous tangential centers in Filippov vector fields

TL;DR

This paper proves that tangential centers in planar Filippov vector fields cannot be isochronous, using analysis of the period function to establish the non-existence of such centers.

Contribution

It provides a novel result showing the impossibility of isochronous tangential centers in Filippov systems, expanding understanding of their qualitative behavior.

Findings

Tangential centers in Filippov vector fields are never isochronous.

The period function analysis confirms non-existence of isochronous centers.

This result clarifies the structure of Filippov systems near tangential centers.

Abstract

The isochronicity problem is a classical problem in the qualitative theory of planar vector fields which consists in characterizing whether a center is isochronous or not, that is, if all the trajectories in a neighbourhood of the center have the same period. This problem is usually investigated by means of the so-called period function. In this paper, we are interested in exploring the isochronicity problem for tangential centers of planar Filippov vector fields. By computing the period function for planar Filippov vector fields around tangential centers, we show that such centers are never isochronous.