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

Journal of Dynamics and Differential Equations

Volumn 25, Issue 4, 2013, Pages 925-958

  Broer, Henk W ^a   Kaper, Tasso J ^b   Krupa, Martin ^c  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

^b Boston University   (United States)

^c INRIA   (France)

Author keywords

Cusp; Geometric desingularization; Geometric singular perturbation theory; Slow fast systems; Takens' classification of singularities; Zeeman's nerve impulse example

Indexed keywords

EID: 84890428688     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1007/s10884-013-9322-5     Document Type: Article

References (19)

1. 3 et leur canards. Asterisque 109-110, 159-191 (1983).
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).
3. Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), 1-100 (1996).
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).
5. Dumortier, F., Roussarie, R., Sotomayor, J.: Bifurcations of cuspidal loops. Nonlinearity 10, 1369-1408 (1997).
6. Fenichel, N.: Geometric singular perturbation theory. J. Differ. Equ. 31, 53-98 (1979).
7. Hirsch, M. W., Pugh, C. C., Shub, M.: Invariant Manifolds, LNM, vol. 583. Springer, Berlin (1977).
8. Jones, C. K. R. T.: Geometric singular perturbation theory. In: Dynamical Systems. LNM vol. 1609. Springer, Berlin (1995).
9. Keener, J., Sneyd, J.: Mathematical Physiology. Interdisciplinary Applied Mathematics, vol. 8. Springer, New York (1998).
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).
11. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174, 312-368 (2001).
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).
13. 3. J. Differ. Equ. 177(2), 419-453 (2001).
14. 3. J. Differ. Equ. 200, 69-104 (2004).
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).
16. Thom, R.: Ensembles et Morphismes Stratifies. Bull. Am. Math. Soc. 75, 240-284 (1969).
17. Thom, R.: L'evolution temporelle de catastrophes. In: Applications of Global Analysis I. Rijksuniversiteit Utrecht, Utrecht (1974).
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).
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).