The discontinuous planar piecewise linear system with two nodes has at most two limit cycles

TL;DR

This paper proves that a specific class of planar piecewise linear systems with two nodes can have at most two limit cycles, using Poincare maps and successor function analysis.

Contribution

It introduces a novel analytical approach to determine the maximum number of limit cycles in piecewise linear systems with two nodes.

Findings

The system has at most two limit cycles.

The method involves constructing Poincare half maps and analyzing a successor function.

The approach provides a bound on limit cycle multiplicity.

Abstract

This paper investigates the multiplicity and the number of limit cycles for planar piecewise linear system divided into two regions by a straight line and each linear subsystem has a node. Through constructing Poincare half maps and a successor function, and analyzing the properties of the successor function, we can derive that this system has at most two limit cycles, counting the multiplicities of limit cycles.