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.
Bibliographical noteFunding Information:
\ast Received by the editors May 26, 2020; accepted for publication (in revised form) April 26, 2021; published electronically June 21, 2021. https://doi.org/10.1137/20M1340113 Funding: The work of the first author was supported by National Science Foundation grant DMS-2016216 (formerly DMS-1815315). The work of the second author was supported by the German Research Fund (DFG) through grant RA 2788/1-1. The work of the third author was supported by National Science Foundation grant DMS-1714429. \dagger School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (pcarter@ umn.edu). \ddagger Fachbereich Mathematik, University of Bremen, 28359, Bremen, Germany (firstname.lastname@example.org). \S Division of Applied Mathematics, Brown University, Providence, RI 02912 USA (Bjorn Sandstede@brown.edu).
© 2021 Society for Industrial and Applied Mathematics.
- Absolute spectrum
- FitzHugh-Nagumo equation
- Geometric singular perturbation theory
- Spectral stability
- Traveling pulses