Fast and slow waves in the FitzHugh-Nagumo equation

Journal of Differential Equations

Volumn 133, Issue 1, 1997, Pages 49-97

  Krupa, Martin ^a   Sandstede, Björn ^b   Szmolyan, Peter ^a  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^b WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030637321     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1996.3198     Document Type: Article

References (31)

1. 1. P. Brunovsky, Tracking invariant manifolds without differential forms, Acta Univ. Comeniana 65 (1996), 23-32.
2. 2. G. Carpenter, A geometric approach to singular perturbation problems with application to nerve impulse equations, J. Differential Equations 23 (1977), 152-173.
3. 3. A. R. Champneys, J. Härterich, and B. Sandstede, A non-transverse homoclinic orbit to a saddle-node equilibrium, Erg. Th. and Dyn. Sys. 16 (1996), 431-450.
4. 4. S.-N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Springer, New York, 1982.
5. 5. S.-N. Chow, B. Deng, and B. Fiedler, Homoclinic Bifurcations at resonant eigenvalues, Dyn. Differential Equations 2 (1990), 177-244.
6. 1-linearization, J. Differential Equations 79 (1989), 189-231.
7. 7. B. Deng, Homoclinic bifurcations with non-hyperbolic equilibria, SIAM J. Math. Anal. 21 (1990), 693-720.
8. 8. B. Deng, The existence of infinitely many travelling front and back waves in the FitzHugh-Nagumo equations, SIAM J. Math. Anal. 22 (1991), 1631-1650.
9. 9. G. Flores, Stability analysis for the slow travelling pulse of the FitzHugh-Nagumo system, SIAM J. Math. Anal. 22 (1991), 392-399.
10. 10. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), 53-98.
11. 11. S. Hastings, On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations, Quart. J. Math. Oxford Ser. (2) 27 (1976), 123-134.
12. 12. A. J. Homburg, H. Kokubu, and M. Krupa, The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit, Erg. Th. and Dyn. Sys. 14 (1994), 667-693.
13. 13. C. K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc. 286 (1984), 431-469.
14. 14. C. K. R. T. Jones, N. Kopell, and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, in "Patterns and Dynamics in reactive Media" (H. Swinney, G. Aris, and D. G. Aronson, Eds.), pp. 101-116, IMA Volumes in Mathematics and its Applications, Vol. 37, Springer, New York, 1991.
15. 15. C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularity perturbed systems, J. Differential Equations 108 (1994), 64-88.
16. 16. C. K. R. T. Jones, N. Kopell, and T. J. Kaper, tracking invariant manifolds up to exponentially small errors, SIAM J. Math. Anal. 27 (1996), 558-577.
17. 17. H. Kokubu, Y. Nishiura, and H. Oka, Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems, J. Differential Equations 86 (1990), 260-341.
18. 18. X.-B. Lin, Using Melnikov method to solve Shilnikov problems, Proc. Roy. Soc. Edinburgh Ser. A 116 (1990), 295-325.
19. 19. R. Langer, "Existence of Homoclinic Travelling Wave Solutions to the FitzHugh-Nagumo Equations," Ph.D. thesis, Northwestern Univ., Evanston, IL, 1980.
20. 20. V. Naudot, Strange attractor in the unfolding of an inclination-flip homoclinic orbit, Erg. Th. and Dyn. Sys., to appear.
21. 21. S. Nii, N-homoclinic bifurcations for homoclinic orbits changing its twisting, J. Dyn. Differential Equations, to appear.
22. 22. B. Sandstede, "Verzweigungstheorie homokliner Verdopplungen," Report No. 7, Institut für Angewandte Analysis und Stochastik, Berlin, 1993.
23. 23. B. Sandstede, Exchange of stability of the fast and slow waves in the FitzHugh-Nagumo equation, preprint.
24. 24. B. Sandstede, A unified approach to homoclinic bifurcation with codimension 2: II, Applications, preprint.
25. 25. B. Sandstede, Numerical computation of homoclinic flip-birfurcations, in preparation.
26. 26. P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations 92 (1991), 252-281.
27. 27. P. Szmolyan, A proof of the Exchange Lemma using Shilnikov coordinates, preprint, Technical Univ. Vienna, 1995.
28. 28. S.-K. Tin, "On the Dynamics of Tangent Spaces near a Normally Hyperbolic Invariant Manifold," Ph.D. thesis, Brown Univ., Providence, RI, 1994.
29. 29. S.-K. Tin, N. Kopell, and C. K. R. T. Jones, Invariant manifolds and singularity perturbed boundary value problems, SIAM J. Numer. Anal. 31 (1994), 1558-1576.
30. 30. S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods," Springer, New York, 1988.
31. 31. E. Yanagida, Stability of fast travelling wave solutions of the FitzHugh-Nagumo equations, J. Math. Biol. 22 (1985), 81-104.