Crossing limit cycles of discontinuous piecewise differential systems with Pleshkan's isochronous centers

TL;DR

This paper investigates the maximum number of crossing limit cycles in planar discontinuous piecewise differential systems with linear and cubic isochronous centers, providing explicit bounds and examples.

Contribution

It establishes explicit upper bounds for crossing limit cycles in fifteen classes of systems and constructs examples with three crossing limit cycles, extending previous results.

Findings

Explicit upper bounds for crossing limit cycles in most configurations

Construction of examples with three crossing limit cycles in each class

Identification of three configurations with open bounds

Abstract

In recent decades, piecewise linear differential systems have attracted considerable attention due to their ability to describe a wide range of phenomena. A central problem, as in the theory of general planar differential systems, is to determine the existence and the maximal number of crossing limit cycles. However, deriving sharp upper bounds for this quantity remains a highly challenging problem. In this work we study crossing limit cycles in planar discontinuous piecewise differential systems separated by a straight line, where each subsystem is either a linear center or a cubic isochronous center with homogeneous nonlinearities. Within this setting, we consider all possible combinations arising from these families, leading to fifteen distinct classes of piecewise systems. Using the existence of first integrals, we reduce the detection of crossing limit cycles to algebraic closing conditions on the discontinuity set, which allows for a systematic and unified analysis across all configurations. As a consequence, we establish explicit upper bounds for the number of crossing limit cycles in all cases except for three configurations that remain open. In addition, we construct examples exhibiting three crossing limit cycles in every class, providing a nontrivial uniform lower bound. Our results extend and complement earlier work in the literature by including previously unstudied configurations and improving some known bounds, thereby providing a comprehensive description of the number of crossing limit cycles within this class of systems