An unbounded number of canard limit cycles in linear regularizations of piecewise linear systems

TL;DR

This paper demonstrates that linear regularizations of piecewise linear systems can have an arbitrarily large finite number of canard limit cycles, using novel breaking mechanisms and the slow divergence integral.

Contribution

It introduces new methods to construct systems with any finite number of hyperbolic limit cycles in piecewise linear systems.

Findings

Constructed systems with arbitrary finite limit cycles.

Introduced Hopf and jump breaking mechanisms.

Used slow divergence integral for analysis.

Abstract

The purpose of this paper is to study the number of limit cycles of canard type in linear regularizations of piecewise linear systems with non-monotonic transition functions. Using the notion of slow divergence integral and elementary breaking mechanisms, we construct systems with an arbitrary finite number of hyperbolic limit cycles. The Hopf breaking mechanism deals with transition functions with precisely one critical point in the interval $(-1,1)$. On the other hand, the jump breaking mechanism produces any number of limit cycles using transition functions with precisely three critical points in $(-1,1)$.