Canonical models for torus canards in elliptic bursters

E. Baspinar*, D. Avitabile, M. Desroches

*Corresponding author for this work

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Abstract

We revisit elliptic bursting dynamics from the viewpoint of torus canard solutions. We show that at the transition to and from elliptic burstings, classical or mixed-type torus canards may appear, the difference between the two being the fast subsystem bifurcation that they approach: saddle-node of cycles for the former and subcritical Hopf for the latter. We first showcase such dynamics in a Wilson-Cowan-type elliptic bursting model, then we consider minimal models for elliptic bursters in view of finding transitions to and from bursting solutions via both kinds of torus canards. We first consider the canonical model proposed by Izhikevich [SIAM J. Appl. Math. 60, 503-535 (2000)] and adapted to elliptic bursting by Ju et al. [Chaos 28, 106317 (2018)] and we show that it does not produce mixed-type torus canards due to a nongeneric transition at one end of the bursting regime. We, therefore, introduce a perturbative term in the slow equation, which extends this canonical form to a new one that we call Leidenator and which supports the right transitions to and from elliptic bursting via classical and mixed-type torus canards, respectively. Throughout the study, we use singular flows (ε=0) to predict the full system’s dynamics (ε>0 small enough). We consider three singular flows, slow, fast, and average slow, so as to appropriately construct singular orbits corresponding to all relevant dynamics pertaining to elliptic bursting and torus canards. Finally, we comment on possible links with mixed-type torus canards and folded-saddle-node singularities in non-canonical elliptic bursters that possess a natural three-timescale structure.

Original languageEnglish
Article number063129
Pages (from-to)1-15
Number of pages15
JournalChaos
Volume31
Issue number6
Early online date15 Jun 2021
DOIs
Publication statusPublished - Jun 2021

Bibliographical note

Publisher Copyright:
© 2021 Author(s).

Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

Funding

FundersFunder number
Horizon 2020 Framework Programme785907

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