Canards Existence in the Hindmarsh–Rose Model

Jean Marc Ginoux, Jaume Llibre, Kiyoyuki Tchizawa

Research output: Chapter in BookChapterResearchpeer-review

Abstract

© 2019, Springer Nature Switzerland AG. 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; see [4, 5]. 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
Title of host publicationTrends in Mathematics
Pages169-175
Number of pages6
Volume11
ISBN (Electronic)2297-024X
DOIs
Publication statusPublished - 1 Jan 2019

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