| Issue |
Math. Model. Nat. Phenom.
Volume 14, Number 4, 2019
Singular perturbations and multiscale systems
|
|
|---|---|---|
| Article Number | 406 | |
| Number of page(s) | 14 | |
| DOI | https://doi.org/10.1051/mmnp/2019018 | |
| Published online | 28 May 2019 | |
Parabolic bursting, spike-adding, dips and slices in a minimal model*
1
MathNeuro Team, Inria Sophia Antipolis – Méditerranée,
Sophia Antipolis, France.
2
Université Côte d’Azur,
Nice, France.
3
Laboratoire Jacques-Louis Lions, Sorbonne Université,
Paris, France.
4
Université de Nice Sophia Antipolis, Laboratoire J.-A. Dieudonné,
Nice, France.
** Corresponding author: mathieu.desroches@inria.fr
Received:
26
October
2018
Accepted:
24
April
2019
A minimal system for parabolic bursting, whose associated slow flow is integrable, is presented and studied both from the viewpoint of bifurcation theory of slow-fast systems, of the qualitative analysis of its phase portrait and of numerical simulations. We focus the analysis on the spike-adding phenomenon. After a reduction to a periodically forced one-dimensional system, we uncover the link with the dips and slices first discussed by J.E. Littlewood in his famous articles on the periodically forced van der Pol system.
Mathematics Subject Classification: 37C10 / 37C27 / 34C23 / 37G15 / 34C15 / 34E17
Key words: Parabolic bursting / spike-adding / Littlewood’s dips and slices
© The authors. Published by EDP Sciences, 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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