Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region

TL;DR

This paper proves that planar piecewise linear differential systems without sliding regions can have at most one limit cycle, which, if it exists, is hyperbolic and its stability depends on system parameters, using a novel integral approach.

Contribution

The paper introduces a new integral characterization of Poincaré half-maps to establish a sharp upper bound of one limit cycle without spectral case distinctions.

Findings

Maximum of one limit cycle in the system

Limit cycle, if present, is hyperbolic

Stability determined by simple parameter condition

Abstract

In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining the maximum number of limit cycles that these differential systems can have. This problem has been usually approached via large case-by-case analyses which distinguish the many different possibilities for the spectra of the matrices of the differential systems. Here, by using a novel integral characterization of Poincar\'e half-maps, we prove, without unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for the number of limit cycles of these differential systems is one. In addition, it is proven that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the parameters of the system. As a byproduct of our analysis, a condition for the existence of the limit cycle is also derived.