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

Renato Huzak, Otavio Henrique Perez

TL;DR

The paper addresses whether linear regularizations of planar piecewise-linear systems with non-monotone transition functions can yield arbitrarily many canard-type limit cycles. It introduces two generic breaking mechanisms—Hopf breaking (one critical point in $(-1,1)$) and jump breaking (three critical points)—and uses the slow divergence integral to translate zeros into hyperbolic cycles. By constructing X,Y and families of transition functions φ_k (with one critical point) and φ_{b,k} (with three critical points), the authors prove that for each target count $k$, there exist regularization parameters and small ε>0 such that at least $k+1$ hyperbolic canard cycles appear inside the regularization stripe. The work advances the understanding of unbounded canard cycles in linear regularizations of PWL systems and connects slow-fast theory, Liénard reductions, and explicit transition-function design, with implications for Hilbert-type problems in slow-fast dynamics.

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 $(-1,1)$.

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

TL;DR

The paper addresses whether linear regularizations of planar piecewise-linear systems with non-monotone transition functions can yield arbitrarily many canard-type limit cycles. It introduces two generic breaking mechanisms—Hopf breaking (one critical point in ) and jump breaking (three critical points)—and uses the slow divergence integral to translate zeros into hyperbolic cycles. By constructing X,Y and families of transition functions φ_k (with one critical point) and φ_{b,k} (with three critical points), the authors prove that for each target count , there exist regularization parameters and small ε>0 such that at least hyperbolic canard cycles appear inside the regularization stripe. The work advances the understanding of unbounded canard cycles in linear regularizations of PWL systems and connects slow-fast theory, Liénard reductions, and explicit transition-function design, with implications for Hilbert-type problems in slow-fast dynamics.

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 . On the other hand, the jump breaking mechanism produces any number of limit cycles using transition functions with precisely three critical points in .
Paper Structure (15 sections, 6 theorems, 66 equations, 7 figures)

This paper contains 15 sections, 6 theorems, 66 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Generic breaking mechanisms
  4. The Hopf breaking mechanism
  5. The jump breaking mechanism
  6. Regularized PWL systems and generic breaking mechanisms
  7. Proof of the main results
  8. Proof of Theorem \ref{['thm-main-Hopf']}
  9. Construction of the transition function
  10. Proof of Theorem \ref{['thm-main-jump']}
  11. Regularity of the breaking parameter
  12. Asymptotics of the fast relation function
  13. Asymptotics of the slow divergence integral
  14. Constructing zeros of the slow divergence integral
  15. Constructing the transition function

Key Result

Theorem A

There exist linear vector fields $X(\cdot,\alpha)$ and $Y(\cdot,\alpha)$, depending on a parameter $\alpha\in \mathbb R$, such that the following is true: for any integer $k>0$, there exist a non-monotonic transition function $\varphi_k:\mathbb{R}\rightarrow\mathbb{R}$, with precisely $1$ critical p

Figures (7)

  • Figure 1: Possible canard cycles in \ref{['LINREG-rescaled']} for $\varepsilon=0$. (a) Canard cycle with a slow-fast Hopf point. (b) Canard cycle containing a jump connection.
  • Figure 2: A canard cycle $\Gamma^H_x$ created by the Hopf mechanism.
  • Figure 3: Jump connections.
  • Figure 4: A canard cycle $\Gamma^J_x$ created by the jump mechanism, for $b=0$.
  • Figure 5: A transition function $\varphi_k$ with one critical point in $(-1,1)$, located at $x=0$.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Theorem A: Hopf breaking mechanism
  • Theorem B: Jump breaking mechanism
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Remark 1
  • Lemma 3.4
  • proof
  • Remark 2
  • ...and 1 more