Issue |
Math. Model. Nat. Phenom.
Volume 14, Number 4, 2019
Singular perturbations and multiscale systems
|
|
---|---|---|
Article Number | 409 | |
Number of page(s) | 21 | |
DOI | https://doi.org/10.1051/mmnp/2019012 | |
Published online | 12 July 2019 |
Regular Article
Canards Existence in the Hindmarsh–Rose model
1
Laboratoire d’Informatique et des Systèmes, UMR, CNRS 7020, Université de Toulon, BP 20132,
83957
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: jllibre@mat.uab.cat
Received:
23
October
2018
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.