Unpeeling a homoclinic banana in the fitzhugh–nagumo system

Paul Carter; Björn Sandstede

doi:10.1137/16M1080707

Unpeeling a homoclinic banana in the fitzhugh–nagumo system

Paul Carter, Björn Sandstede

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

Abstract

The FitzHugh–Nagumo equations are known to admit fast traveling pulse solutions with monotone tails. It is also known that this system admits traveling pulses with exponentially decaying oscillatory tails. Upon numerical continuation in parameter space, it has been observed that the oscillations in the tails of the pulses grow into a secondary excursion resembling a second copy of the primary pulse. In this paper, we outline in detail the geometric mechanism responsible for this single-to-double-pulse transition, and we construct the transition analytically using geometric singular perturbation theory and blow-up techniques.

Original language	English (US)
Pages (from-to)	236-349
Number of pages	114
Journal	SIAM Journal on Applied Dynamical Systems
Volume	17
Issue number	1
DOIs	https://doi.org/10.1137/16M1080707
State	Published - 2018
Externally published	Yes

Bibliographical note

Funding Information:
∗Received by the editors June 20, 2016; accepted for publication (in revised form) December 1, 2017; published electronically January 30, 2018. http://www.siam.org/journals/siads/17-1/M108070.html Funding: The work of the first author was supported by the NSF under grant DMS-1148284. The work of the second author was partially supported by the NSF through grant DMS-1408742. †Department of Mathematics, University of Arizona, Tucson, AZ 85721 (pacarter@math.arizona.edu). ‡Division of Applied Mathematics, Brown University, Providence, RI 02912 (Bjorn Sandstede@brown.edu).

Funding Information:
The work of the first author was supported by the NSF under grant DMS-1148284. The work of the second author was partially supported by the NSF through grant DMS-1408742. The authors thank the referees and editor for their constructive and detailed comments. The first author would like to also thank Martin Wechselberger for helpful discussions during an initial stage of this project.

Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

Keywords

Blow-up
Canards
FitzHugh–Nagumo
Geometric singular perturbation theory
Traveling waves

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abstract = "The FitzHugh–Nagumo equations are known to admit fast traveling pulse solutions with monotone tails. It is also known that this system admits traveling pulses with exponentially decaying oscillatory tails. Upon numerical continuation in parameter space, it has been observed that the oscillations in the tails of the pulses grow into a secondary excursion resembling a second copy of the primary pulse. In this paper, we outline in detail the geometric mechanism responsible for this single-to-double-pulse transition, and we construct the transition analytically using geometric singular perturbation theory and blow-up techniques.",

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Unpeeling a homoclinic banana in the fitzhugh–nagumo system

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