Motivated by pulse-replication phenomena observed in the FitzHugh--Nagumo equation, we investigate traveling pulses whose slow/fast profiles exhibit canard-like transitions. We show that the spectra of the PDE linearization about such pulses may contain many point eigenvalues that accumulate onto a union of curves as the slow scale parameter approaches zero. The limit sets are related to the absolute spectrum of the homogeneous rest states involved in the canard-like transitions. Our results are formulated for general systems that admit an appropriate slow/fast structure.


  1. spectral stability
  2. traveling pulses
  3. FitzHugh--Nagumo equation
  4. geometric singular perturbation theory
  5. absolute spectrum
  6. canards

MSC codes

  1. 35B35
  2. 35P15
  3. 35C07
  4. 35B25
  5. 37L15
  6. 34E17

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 3520 - 3576
ISSN (online): 1095-7154


Submitted: 26 May 2020
Accepted: 26 April 2021
Published online: 21 June 2021


  1. spectral stability
  2. traveling pulses
  3. FitzHugh--Nagumo equation
  4. geometric singular perturbation theory
  5. absolute spectrum
  6. canards

MSC codes

  1. 35B35
  2. 35P15
  3. 35C07
  4. 35B25
  5. 37L15
  6. 34E17



Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : RA 2788/1-1

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS-1714429

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS-2016216

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