Four Obsessions of the Two-Fold Singularity

  • Mike R. Jeffrey


And so we return, one last time, to the two-fold singularity. The story of this elementary point spans the last 30 years of piecewise-smooth dynamical theory. No other aspect of nonsmooth systems has so challenged the established wisdom or tested and inspired new ideas.


  1. 4.
    M. A. Aizerman and F. R. Gantmakher. On the stability of equilibrium positions in discontinuous systems. Prikl. Mat. i Mekh., 24:283–93, 1960.zbMATHGoogle Scholar
  2. 37.
    A. Colombo and M. R. Jeffrey. Non-deterministic chaos, and the two-fold singularity in piecewise smooth flows. SIAM J. App. Dyn. Sys., 10:423–451, 2011.zbMATHGoogle Scholar
  3. 38.
    A. Colombo and M. R. Jeffrey. The two-fold singularity: leading order dynamics in n-dimensions. Physica D, 263:1–10, 2013.MathSciNetCrossRefGoogle Scholar
  4. 44.
    M. Desroches and M. R. Jeffrey. Canards and curvature: nonsmooth approximation by pinching. Nonlinearity, 24:1655–1682, 2011.MathSciNetzbMATHGoogle Scholar
  5. 45.
    M. Desroches and M. R. Jeffrey. Nonsmooth analogues of slow-fast dynamics: pinching at a folded node. arXiv:1506.00831, 2013.Google Scholar
  6. 47.
    M. Desroches, B. Krauskopf, and H. M. Osinga. Numerical continuation of canard orbits in slow-fast dynamical systems. Nonlinearity, 23(3):739–765, 2010.MathSciNetzbMATHGoogle Scholar
  7. 56.
    F. Dumortier, R. Roussarie, J. Sotomayor, and H. Zoladek. Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals. Springer-Verlag, 1991.Google Scholar
  8. 69.
    S. Fernández-Garcia, D. Angulo-Garcia, G. Olivar-Tost, M. di Bernardo, and M. R. Jeffrey. Structural stability of the two-fold singularity. SIAM J. App. Dyn. Sys., 11(4):1215–1230, 2012.MathSciNetzbMATHGoogle Scholar
  9. 70.
    A. F. Filippov. Differential equations with discontinuous right-hand side. American Mathematical Society Translations, Series 2, 42:19–231, 1964.Google Scholar
  10. 71.
    A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publ. Dortrecht, 1988 (Russian 1985).Google Scholar
  11. 116.
    M. R. Jeffrey. Hidden degeneracies in piecewise smooth dynamical systems. Int. J. Bif. Chaos, 26(5):1650087(1–18), 2016.Google Scholar
  12. 117.
    M. R. Jeffrey. The ghosts of departed quantities in switches and transitions. SIAM Review, 60(1):116–36, 2017.MathSciNetzbMATHGoogle Scholar
  13. 119.
    M. R. Jeffrey and A. Colombo. The two-fold singularity of discontinuous vector fields. SIAM Journal on Applied Dynamical Systems, 8(2):624–640, 2009.MathSciNetCrossRefGoogle Scholar
  14. 123.
    Mike R. Jeffrey. An update on that singularity. Trends in Mathematics: Research Perspectives CRM Barcelona (Birkhauser), 8:107–122, 2017.MathSciNetGoogle Scholar
  15. 133.
    K. U. Kristiansen and S. J. Hogan. On the use of blowup to study regularization of singularities of piecewise smooth dynamical systems in R3. SIADS, 14(1):382–422, 2015.zbMATHGoogle Scholar
  16. 138.
    Y. A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, 3rd Ed., 2004.Google Scholar
  17. 139.
    Y. A. Kuznetsov. Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcations. Int. J. Bif. Chaos, 15:3535–46, 2005.MathSciNetzbMATHGoogle Scholar
  18. 140.
    Yu. A. Kuznetsov, S. Rinaldi, and A. Gragnani. One-parameter bifurcations in planar Filippov systems. Int. J. Bif. Chaos, 13:2157–2188, 2003.MathSciNetCrossRefGoogle Scholar
  19. 151.
    J. Llibre, P. R. da Silva, and M. A. Teixeira. Sliding vector fields via slow-fast systems. Bull. Belg. Math. Soc. Simon Stevin, 15(5):851–869, 2008.MathSciNetzbMATHGoogle Scholar
  20. 166.
    Yu. I. Neimark and S. D. Kinyapin. On the equilibrium state on a surface of discontinuity. Izv. VUZ. Radiofizika, 3:694–705, 1960.Google Scholar
  21. 202.
    F. Takens. Forced oscillations and bifurcations. Comm. Math. Inst. Rijksuniv. Utrecht 2, pages 1–111, 1974.MathSciNetzbMATHGoogle Scholar
  22. 205.
    M. A. Teixeira. Stability conditions for discontinuous vector fields. J. Differ. Equ., 88:15–29, 1990.MathSciNetCrossRefGoogle Scholar
  23. 206.
    M. A. Teixeira. Generic bifurcation of sliding vector fields. J. Math. Anal. Appl., 176:436–457, 1993.MathSciNetCrossRefGoogle Scholar
  24. 218.
    M. Wechselberger. Existence and bifurcation of canards in 3 in the case of a folded node. SIAM J. App. Dyn. Sys., 4(1):101–139, 2005.MathSciNetzbMATHGoogle Scholar
  25. 219.
    M. Wechselberger. A propos de canards (apropos canards). Trans. Amer. Math. Soc, 364:3289–3309, 2012.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mike R. Jeffrey
    • 1
  1. 1.Department of Engineering MathematicsUniversity of BristolBristolUK

Personalised recommendations