Limit cycles for classes of piecewise smooth differential equations separated by the unit circle

TL;DR

This paper investigates the existence and maximum number of limit cycles in piecewise smooth differential equations separated by the unit circle, including cases with singularities and homoclinic cycles.

Contribution

It provides upper bounds for the number of limit cycles and demonstrates that these bounds can be achieved in specific cases.

Findings

Established an upper bound for the number of limit cycles.

Constructed examples reaching the maximum number of limit cycles.

Discussed the existence of homoclinic cycles in saddle-center cases.

Abstract

In this article we study the existence of limit cycles in families of piecewise smooth differential equations having the unit circle as discontinuity region. We consider families presenting singularities of center or saddle type, visible or invisible, as well as the case without singularities. We establish an upper bound for the number of limit cycles and give examples showing that the maximum number of limit cycles can be reached. We also discuss the existence of homoclinic cycles for such differential equations in the saddle-center case.