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Journal of Nonlinear Science

, Volume 26, Issue 5, pp 1369–1444 | Cite as

Stability of Traveling Pulses with Oscillatory Tails in the FitzHugh–Nagumo System

  • Paul CarterEmail author
  • Björn de Rijk
  • Björn Sandstede
Article

Abstract

The FitzHugh–Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in an appropriate singular limit, where a parameter \(\varepsilon \) goes to zero. These pulses are known to be nonlinearly stable with respect to the underlying PDE. Recently, the existence of fast pulses with oscillatory tails was proved for the FitzHugh–Nagumo equations. In this paper, we prove that the fast pulses with oscillatory tails are also nonlinearly stable. Similar to the case of monotone tails, stability is decided by the location of a nontrivial eigenvalue near the origin of the PDE linearization about the traveling pulse. We prove that this real eigenvalue is always negative. However, the expression that governs the sign of this eigenvalue for oscillatory pulses differs from that for monotone pulses, and we show indeed that the nontrivial eigenvalue in the monotone case scales with \(\varepsilon \), while the relevant scaling in the oscillatory case is \(\varepsilon ^{2/3}\).

Keywords

FitzHugh-Nagumo system traveling pulses spectral stability geometric singular perturbation theory Lin’s method 

Mathematics Subject Classification

35B35 35C07 35B25 35P15 35K57 

Notes

Acknowledgments

Carter was supported by the NSF under Grant DMS-1148284. De Rijk was supported by the Dutch science foundation (NWO) cluster NDNS+. Sandstede was partially supported by the NSF through Grant DMS-1409742.

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Beck, M., Jones, C.K.R.T., Schaeffer, D., Wechselberger, M.: Electrical waves in a one-dimensional model of cardiac tissue. SIAM J. Appl. Dyn. Syst. 7(4), 1558–1581 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Carpenter, G.: A geometric approach to singular perturbation problems with applications to nerve impulse equations. J. Differ. Equ. 23(3), 335–367 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Carter, P., Sandstede, B.: Fast pulses with oscillatory tails in the FitzHugh–Nagumo system. SIAM J. Math. Anal. 47(5), 3393–3441 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Coppel, W.A.: Dichotomies and reducibility. J. Differ. Equ. 3, 500–521 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Coppel, W.A.: Dichotomies in Stability Theory, Volume 629 of Lecture Notes in Mathematics. Springer, Berlin (1978)CrossRefGoogle Scholar
  6. Eszter, E.G.: An Evans function analysis of the stability of periodic travelling wave solutions of the FitzHugh–Nagumo system. PhD thesis, University of Massachusetts (1999)Google Scholar
  7. Evans, J.W.: Nerve axon equations. I. Linear approximations. Indiana Univ. Math. J. 21, 877–885 (1971/1972)Google Scholar
  8. Evans, J.W.: Nerve axon equations. III. Stability of the nerve impulse. Indiana Univ. Math. J. 22, 577–593 (1972/1973)Google Scholar
  9. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  10. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1(6), 445 (1961)CrossRefGoogle Scholar
  11. Flores, G.: Stability analysis for the slow travelling pulse of the FitzHugh–Nagumo system. SIAM J. Math. Anal. 22(2), 392–399 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hastings, S.P.: On the existence of homoclinic and periodic orbits for the FitzHugh–Nagumo equations. Q. J. Math. 27(1), 123–134 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hastings, S.P.: Single and multiple pulse waves for the Fitzhugh–Nagumo. SIAM J. Appl. Math. 42(2), 247–260 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Holzer, M., Doelman, A., Kaper, T.J.: Existence and stability of traveling pulses in a reaction–diffusion-mechanics system. J. Nonlinear Sci. 23(1), 129–177 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Homburg, A.J., & Sandstede, B.: Homoclinic and heteroclinic bifurcations in vector fields. In: Broer, H., Takens, F., Hasselblatt, B. (eds.) Handbook of Dynamical Systems III, pp 379–524. Elsevier (2010)Google Scholar
  16. Hupkes, H.J., Sandstede, B.: Stability of pulse solutions for the discrete FitzHugh–Nagumo system. Trans. Am. Math. Soc. 365(1), 251–301 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Jones, C.K.R.T.: Stability of the travelling wave solution of the FitzHugh–Nagumo system. Trans. Am. Math. Soc. 286(2), 431–469 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Jones, C.K.R.T., Kaper, T.J., Kopell, N.: Tracking invariant manifolds up to exponentially small errors. SIAM J. Math. Anal. 27(2), 558–577 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Jones, C.K.R.T., Kopell, N., Langer, R.: Construction of the FitzHugh–Nagumo pulse using differential forms. In: Patterns and Dynamics in Reactive Media, pp. 101–115. Springer (1991)Google Scholar
  20. Kapitula, T., Promislow, K.: Spectral and Dynamical Stability of Nonlinear Waves, volume 185 of Applied Mathematical Sciences. Springer, New York (2013). With a foreword by C. K. R. T JonesGoogle Scholar
  21. Krupa, M., Sandstede, B., Szmolyan, P.: Fast and slow waves in the FitzHugh–Nagumo equation. J. Differ. Equ. 133(1), 49–97 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Krupa, M., Szmolyan, P.: Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions. SIAM J. Math. Anal. 33(2), 286–314 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Levinson, N.: The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Lin, X.B.: Using Melnikov’s method to solve Šilnikov’s problems. Proc. R. Soc. Edinb. Sect. A 116(3–4), 295–325 (1990)CrossRefzbMATHGoogle Scholar
  26. Mishchenko, E.: Differential Equations with Small Parameters and Relaxation Oscillations, vol. 13. Springer Science & Business Media, Berlin (2013)Google Scholar
  27. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962)CrossRefGoogle Scholar
  28. Palmer, K.J.: Exponential dichotomies and transversal homoclinic points. J. Differ. Equ. 55(2), 225–256 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Sandstede, B.: Verzweigungstheorie homokliner Verdopplungen. PhD thesis, University of Stuttgart (1993)Google Scholar
  30. Sandstede, B.: Stability of multiple-pulse solutions. Trans. Am. Math. Soc. 350(2), 429–472 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Sandstede, B.: Stability of travelling waves. In: Fiedler, B.(ed.) Handbook of Dynamical Systems, vol. 2, pp. 983–1055. Elsevier (2002)Google Scholar
  32. Sandstede, B., Scheel, A.: Absolute and convective instabilities of waves on unbounded and large bounded domains. Phys. D 145(3–4), 233–277 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Yanagida, E.: Stability of fast travelling pulse solutions of the FitzHugh–Nagumo equations. J. Math. Biol. 22(1), 81–104 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Paul Carter
    • 1
    Email author
  • Björn de Rijk
    • 2
  • Björn Sandstede
    • 3
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.Mathematisch InstituutUniversiteit LeidenLeidenThe Netherlands
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA

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