Math. Model. Nat. Phenom.
Volume 14, Number 4, 2019
Singular perturbations and multiscale systems
|Number of page(s)||21|
|Published online||12 July 2019|
Canards Existence in the Hindmarsh–Rose model
Laboratoire d’Informatique et des Systèmes, UMR, CNRS 7020, Université de Toulon, BP 20132,
La Garde cedex, France.
2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain.
3 Institute of Administration Engineering, Ltd., Tokyo 101-0021, Japan.
* Corresponding author: email@example.com
Accepted: 13 February 2019
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.
Mathematics Subject Classification: 34C23 / 34C25 / 37G10
Key words: Hindmarsh–Rose model / singularly perturbed dynamical systems / canard solutions
© EDP Sciences, 2019
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