Sliding cycles of the regularized piecewise linear $VI_3$ two-fold

TL;DR

This paper investigates the maximum number of sliding limit cycles in a regularized piecewise linear two-fold system, showing that at most two such cycles can occur and identifying parameter regions with two cycles.

Contribution

It introduces the use of the slow divergence integral to analyze sliding limit cycles in a regularized piecewise linear system, establishing an upper bound of two cycles.

Findings

The integral has at most one zero, limiting the number of canard cycles.

Canard cycles can produce up to two limit cycles.

Regions in parameter space with exactly two limit cycles are identified.

Abstract

The goal of this paper is to study the number of sliding limit cycles of a regularized piecewise linear $VI_3$ two-fold using the notion of slow divergence integral. We focus on limit cycles produced by canard cycles located in the half-plane with an invisible fold point. We prove that the integral has at most $1$ zero counting multiplicity (when it is not identically zero). This will imply that the canard cycles can produce at most $2$ limit cycles. Moreover, we detect regions in the parameter space with $2$ limit cycles.