Towards Theory of Piecewise Linear Dynamical Systems

TL;DR

This paper analyzes a class of planar piecewise linear dynamical systems with multiple dropping sections, establishing an upper bound on the number of limit cycles based on the number of sections and singular points.

Contribution

It provides a theoretical upper bound on the number of limit cycles in piecewise linear systems with multiple dropping sections, extending understanding of their bifurcation behavior.

Findings

Maximum of $k+2$ limit cycles for systems with $k$ dropping sections.

$k+1$ limit cycles surround individual foci.

One limit cycle surrounds all singular points.

Abstract

In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary number (but finite) of dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system with $k$ dropping sections and $2k+1$ singular points can have at most $k+2$ limit cycles, $k+1$ of which surround the foci one by one and the last, $(k+2)$-th, limit cycle surrounds all of the singular points of this system.