Canards existence in the Hindmarsh-Rose model
Ginoux, Jean-Marc (Centre National de la Recherche Scientifique (França))
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Tchizawa, Kiyoyuki (Institute of Administration Engineering Ltd. (Japan))

Date:	2019
Abstract:	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.
Grants:	Ministerio de Economía y Competitividad MTM2016-77278-P Ministerio de Economía y Competitividad MTM2013-40998-P
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Hindmarsh-Rose model ; Singularly perturbed dynamical systems ; Canard solutions
Published in:	Mathematical Modelling of Natural Phenomena, Vol. 14, Issue 4 (2019) , art. 409, ISSN 1760-6101

DOI: 10.1051/mmnp/2019012

Postprint 16 p, 706.1 KB

21 p, 529.6 KB

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