Journal of Dynamics and Differential Equations

, Volume 25, Issue 4, pp 925–958 | Cite as

Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples



The cusp singularity—a point at which two curves of fold points meet—is a prototypical example in Takens’ classification of singularities in constrained equations, which also includes folds, folded saddles, folded nodes, among others. In this article, we study cusp singularities in singularly perturbed systems for sufficiently small values of the perturbation parameter, in the regime in which these systems exhibit fast and slow dynamics. Our main result is an analysis of the cusp point using the method of geometric desingularization, also known as the blow-up method, from the field of geometric singular perturbation theory. Our analysis of the cusp singularity was inspired by the nerve impulse example of Zeeman, and we also apply our main theorem to it. Finally, a brief review of geometric singular perturbation theory for the two elementary singularities from the Takens’ classification occurring for the nerve impulse example—folds and folded saddles—is included to make this article self-contained.


Cusp Slow\(-\)fast systems Geometric singular perturbation theory  Geometric desingularization Takens’ classification of singularities  Zeeman’s nerve impulse example 


  1. 1.
    Benoit, E.: Systemes lentes-rapides en \({\mathbb{R}}^3\) et leur canards. Asterisque 109–110, 159–191 (1983)MathSciNetGoogle Scholar
  2. 2.
    Dumortier, F.: Techniques in the theory of local bifurcations: blow-up, normal forms, nilpotent bifurcations, singular perturbations. In: Schlomiuk, D. (ed.) Bifurcations and Periodic Orbits of Vector Fields. NATO ASI Series C, Mathematical and Physical Sciences, vol. 408, pp. 19–73. Kluwer, Dordrecht (1993)Google Scholar
  3. 3.
    Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), 1–100 (1996)MathSciNetGoogle Scholar
  4. 4.
    Dumortier, F., Roussarie, R.: Geometric singular perturbation theory beyond normal hyperbolicity. In: Jones, C.K.R.T., Khibnik, A. (eds.) Multiple-Time-Scale Dynamical Systems. IMA Vol. Math. Appl., vol. 122, pp. 29–63, Springer, New York (2001)Google Scholar
  5. 5.
    Dumortier, F., Roussarie, R., Sotomayor, J.: Bifurcations of cuspidal loops. Nonlinearity 10, 1369–1408 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fenichel, N.: Geometric singular perturbation theory. J. Differ. Equ. 31, 53–98 (1979)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds, LNM, vol. 583. Springer, Berlin (1977)Google Scholar
  8. 8.
    Jones, C.K.R.T.: Geometric singular perturbation theory. In: Dynamical Systems. LNM vol. 1609. Springer, Berlin (1995)Google Scholar
  9. 9.
    Keener, J., Sneyd, J.: Mathematical Physiology. Interdisciplinary Applied Mathematics, vol. 8. Springer, New York (1998)Google Scholar
  10. 10.
    Krupa, M., Szmolyan, P.: Extending geometric singular perturbation theory to nonhyperbolic points: fold and canard points in two dimensions. SIAM J. Math. Anal. 33, 286–314 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174, 312–368 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Rotstein, H., Wechselberger, M., Kopell, N.: Canard induced mixed-mode oscillations in a medial entorhinal cortex layer II stellate cell model. SIAM J. Appl. Dyn. Syst. 7, 1582–1611 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Szmolyan, P., Wechselberger, M.: Canards in \({\mathbb{R}}^3\). J. Differ. Equ. 177(2), 419–453 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Szmolyan, P., Wechselberger, M.: Relaxation oscillations in \({\mathbb{R}}^3\). J. Differ. Equ. 200, 69–104 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Takens, F.: Constrained equations: a study of implicit differential equations and their discontinuous solutions. In: Structural Stability, the Theory of Catastrophes, and Applications in the Sciences, LNM, vol. 525, pp. 134–234. Springer, Berlin (1976)Google Scholar
  16. 16.
    Thom, R.: Ensembles et Morphismes Stratifies. Bull. Am. Math. Soc. 75, 240–284 (1969)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Thom, R.: L’evolution temporelle de catastrophes. In: Applications of Global Analysis I. Rijksuniversiteit Utrecht, Utrecht (1974)Google Scholar
  18. 18.
    Thom, R.: Structural stability and morphogenesis. An Outline of a General Theory of Models, 2nd edn. Addison-Wesley, Redwood City, CA (1989). (English; French original)Google Scholar
  19. 19.
    Zeeman, E.C.: Differential equations for the heartbeat and nerve impulse. In: Towards a Theoretical Biology, vol. 4, pp. 8–67. Edinburgh University Press (1972)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for MathematicsUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  3. 3.INRIA Paris-Rocquencourt CentreLe Chesnay CedexFrance

Personalised recommendations