Spike-Adding Canard Explosion in a Class of Square-Wave Bursters

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Abstract

This paper examines a spike-adding bifurcation phenomenon whereby small-amplitude canard cycles transition into large-amplitude bursting oscillations along a single continuous branch in parameter space. We consider a class of three-dimensional singularly perturbed ODEs with two fast variables and one slow variable and singular perturbation parameter ε≪ 1 under general assumptions which guarantee such a transition occurs. The primary ingredients include a cubic critical manifold and a saddle homoclinic bifurcation within the associated layer problem. The continuous transition from canard cycles to N-spike bursting oscillations up to N∼ O(1 / ε) spikes occurs upon varying a single bifurcation parameter on an exponentially thin interval. We construct this transition rigorously using geometric singular perturbation theory; critical to understanding this transition are the existence of canard orbits and slow passage through the saddle homoclinic bifurcation, which are analyzed in detail.

Original languageEnglish (US)
Pages (from-to)2613-2669
Number of pages57
JournalJournal of Nonlinear Science
Volume30
Issue number6
DOIs
StatePublished - Dec 1 2020

Bibliographical note

Funding Information:
The author gratefully acknowledges support through NSF Grant DMS-2016216 (formerly DMS-1815315).

Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Bursting oscillations
  • Canards
  • Geometric singular perturbation theory
  • Saddle-homoclinic bifurcation
  • Spike-adding

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