Resolution of the Piecewise Smooth Visible–Invisible Two-Fold Singularity in R3 Using Regularization and Blowup

K. Uldall Kristiansen*, S. J. Hogan

*Corresponding author for this work

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Two-fold singularities in a piecewise smooth (PWS) dynamical system in R3 have long been the subject of intensive investigation. The interest stems from the fact that trajectories which enter the two-fold are associated with forward non-uniqueness. The key questions are: how do we continue orbits forward in time? Are there orbits that are distinguished among all the candidates? We address these questions by regularizing the PWS dynamical system for the case of the visible–invisible two-fold. Within this framework, we consider a regularization function outside the class of Sotomayor and Teixeira. We then undertake a rigorous investigation, using geometric singular perturbation theory and blowup. We show that there is indeed a forward orbit U that is distinguished amongst all the possible forward orbits leaving the two-fold. Working with a normal form of the visible–invisible two-fold, we show that attracting limit cycles can be obtained (due to the contraction towards U), upon composition with a global return mechanism. We provide some illustrative examples.

Original languageEnglish
JournalJournal of Nonlinear Science
Issue number2
Pages (from-to)723-787
Publication statusPublished - 1 Jan 2018


  • Blowup
  • Geometric singular perturbation theory
  • Piecewise smooth systems
  • Regularization
  • Two-fold bifurcation


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