On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line

TL;DR

This paper investigates the maximum number of limit cycles in a planar piecewise linear differential system with two regions separated by a straight line, establishing bounds based on the position of equilibria.

Contribution

It provides new bounds for the maximum number of limit cycles in such systems, especially when an equilibrium lies on the discontinuity line.

Findings

Maximum of 3 limit cycles in the system

At least 2 limit cycles can occur

Bounds depend on equilibrium position

Abstract

In this paper we study the maximum number $N$ of limit cycles that can exhibit a planar piecewise linear differential system formed by two pieces separated by a straight line. More precisely, we prove that this maximum number satisfies $2\leq N \leq 3$ if one of the two linear differential systems has its equilibrium point on the straight line of discontinuity.