WIGGLY CANARDS: GROWTH OF TRAVELING WAVE TRAINS THROUGH A FAMILY OF FAST-SUBSYSTEM FOCI

Paul Carter, Alan R. Champneys

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a canard explosion. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.

Original languageEnglish (US)
Pages (from-to)2433-2466
Number of pages34
JournalDiscrete and Continuous Dynamical Systems - Series S
Volume15
Issue number9
DOIs
StatePublished - Sep 2022

Bibliographical note

Funding Information:
2020 Mathematics Subject Classification. Primary: 34E17; Secondary: 34C23, 37G15, 35C07. Key words and phrases. Canards, traveling waves, periodic orbits, FitzHugh–Nagumo system, geometric singular perturbation theory. The first author is supported by NSF grant DMS-2016216. ∗ Corresponding author: Paul Carter.

Publisher Copyright:
© 2022 American Institute of Mathematical Sciences. All rights reserved.

Keywords

  • Canards
  • FitzHugh-Nagumo system
  • geometric singular perturbation theory
  • periodic orbits
  • traveling waves

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