On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in $\mathbb{R}^3$

TL;DR

This paper applies the blow-up method to analyze regularizations of singularities in 3D piecewise smooth dynamical systems, revealing how regularization affects singular canards and providing numerical evidence for secondary canards.

Contribution

It extends the blow-up technique to study regularizations of singularities in 3D systems, clarifying the persistence of canards and introducing new insights into their behavior.

Findings

Regularization preserves primary singular canards under certain conditions.

A canard exists in the regularized system if a non-resonance condition is met.

Numerical evidence supports the existence of secondary canards near resonance.

Abstract

In this paper we use the blow up method of Dumortier and Roussarie \cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of singularities of piecewise smooth dynamical systems \cite{filippov1988differential} in $\mathbb R^3$. Using the regularization method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the power of our approach by considering the case of a fold line. We quickly recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided a certain non-resonance condition holds. Finally, we provide numerical evidence for the existence of secondary canards near resonance.