Crossing limit cycles of planar discontinuous piecewise differential systems formed by isochronous centers

TL;DR

This paper investigates the maximum number of crossing limit cycles in certain planar discontinuous piecewise differential systems, providing an upper bound and extending the 16th Hilbert problem for these classes.

Contribution

It introduces new bounds for crossing limit cycles in systems combining linear and cubic isochronous centers separated by a line.

Findings

Upper bounds for crossing limit cycles established

Extended the 16th Hilbert problem to specific discontinuous systems

Analyzed systems with linear and cubic isochronous centers

Abstract

These last years an increasing interest appeared for studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. One of the difficulties for understanding the dynamics of these systems is the study their limit cycles. In this paper we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line, and formed by combinations of linear centers (consequently isochronous) and cubic isochronous centers with homogeneous nonlinearities. For these classes of planar discontinuous piecewise differential systems we solved the extension of the 16th Hilbert problem, i.e. we provide an upper bound for their maximum number of crossing limit cycles.