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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stability of pulse solutions for the discrete FitzHugh-Nagumo system


Authors: H. J. Hupkes and B. Sandstede
Journal: Trans. Amer. Math. Soc. 365 (2013), 251-301
MSC (2010): Primary 34A33, 34K26, 34D35, 34K08
Published electronically: July 11, 2012
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Abstract: We show that the fast travelling pulses of the discrete FitzHugh-Nagumo system in the weak-recovery regime are nonlinearly stable. The spectral conditions that need to be verified involve linear operators that are associated to functional differential equations of mixed type. Such equations are ill-posed and do not admit a semi-flow, which precludes the use of standard Evans-function techniques. Instead, we construct the potential eigenfunctions directly by using exponential dichotomies, Fredholm techniques and an infinite-dimensional version of the Exchange Lemma.


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Additional Information

H. J. Hupkes
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
Address at time of publication: Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Email: hjhupkes@dam.brown.edu, hhupkes@math.leidenuniv.nl

B. Sandstede
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
Email: bjorn{\textunderscore}sandstede@brown.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05567-X
PII: S 0002-9947(2012)05567-X
Keywords: Discrete FitzHugh–Nagumo system, travelling pulses, stability, singular perturbation theory, lattice differential equations, functional differential equations.
Received by editor(s): May 6, 2010
Received by editor(s) in revised form: February 1, 2011
Published electronically: July 11, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.