On the use of blowup to study regularizations of singularities of piecewise smooth dynamical systems in R^3

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Abstract

In this paper we use the blowup method of Dumortier and Roussarie, in the formulation due to Krupa and Szmolyan, to study the regularization of singularities of piecewise smooth dynamical systems in R3. Using the regularization method of Sotomayor and Teixeira, we first demonstrate the power of our approach by considering the case of a fold line. We quickly extend a main result of Reves and Seara 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 nonresonance condition holds. Finally, we provide numerical evidence for the existence of secondary canards near resonance.
Original languageEnglish
JournalS I A M Journal on Applied Dynamical Systems
Volume14
Issue number1
Pages (from-to)382–422
ISSN1536-0040
DOIs
Publication statusPublished - 2015

Keywords

  • Piecewise smooth systems
  • Blowup
  • Geometric singular perturbation theory
  • Sliding bifurcations
  • Canards

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