The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly varying quantity replacing a static parameter induces the solutions of the resulting slow–fast system to feel the effect of the Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow–fast dynamical systems; in the present work, we study it for the first time in piecewise linear (PWL) slow–fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular, through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slow–fast system to exhibit a slow passage through a Hopf-like bifurcation, in link with possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi–Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.
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December 2022
Research Article|
December 01 2022
Slow passage through a Hopf-like bifurcation in piecewise linear systems: Application to elliptic bursting
J. Penalva
;
J. Penalva
a)
(Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft)
1
Departament de Matemàtiques i Informàtica & IAC3, Universitat de les Illes Balears
, Palma 07122, Spain
a)Author to whom correspondence should be addressed: [email protected]
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M. Desroches
;
M. Desroches
(Conceptualization, Investigation, Methodology, Supervision, Validation)
2
MathNeuro Project-Team, Inria at Université Côte d’Azur
, Sophia Antipolis 06902, France
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A. E. Teruel
;
A. E. Teruel
(Conceptualization, Investigation, Methodology, Supervision, Validation, Writing – original draft)
1
Departament de Matemàtiques i Informàtica & IAC3, Universitat de les Illes Balears
, Palma 07122, Spain
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C. Vich
C. Vich
(Conceptualization, Investigation, Methodology, Supervision, Validation, Writing – review & editing)
1
Departament de Matemàtiques i Informàtica & IAC3, Universitat de les Illes Balears
, Palma 07122, Spain
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a)Author to whom correspondence should be addressed: [email protected]
Chaos 32, 123109 (2022)
Article history
Received:
June 03 2022
Accepted:
November 09 2022
Citation
J. Penalva, M. Desroches, A. E. Teruel, C. Vich; Slow passage through a Hopf-like bifurcation in piecewise linear systems: Application to elliptic bursting. Chaos 1 December 2022; 32 (12): 123109. https://doi.org/10.1063/5.0101778
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