Journal of Nonlinear Science

, Volume 29, Issue 2, pp 723–787 | Cite as

Resolution of the Piecewise Smooth Visible–Invisible Two-Fold Singularity in \(\mathbb {R}^3\) Using Regularization and Blowup

  • K. Uldall KristiansenEmail author
  • S. J. Hogan


Two-fold singularities in a piecewise smooth (PWS) dynamical system in \(\mathbb {R}^3\) 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.


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

Mathematics Subject Classification

37G10 34E15 37M99 



We would like to thank an anonymous referee whose many suggestions have greatly improved the manuscript.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark
  2. 2.Department of Engineering MathematicsUniversity of BristolBristolUK

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