Canards Existence in the Hindmarsh-Rose model

Jean Marc Ginoux, Jaume Llibre, Kiyoyuki Tchizawa

Research output: Contribution to journalArticleResearch

3 Citations (Scopus)

Abstract

© 2019 EDP Sciences. In two previous papers we have proposed a new method for proving the existence of "canard solutions" on one hand for three and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand for four-dimensional singularly perturbed systems with two fast variables [J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2016) 381-431; J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of "canard solutions" for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of "canard solutions" in the Hindmarsh-Rose model.
Original languageEnglish
Article number409
JournalMathematical Modelling of Natural Phenomena
Volume14
DOIs
Publication statusPublished - 1 Jan 2019

Keywords

  • Canard solutions
  • Hindmarsh-Rose model
  • Singularly perturbed dynamical systems

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