On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter

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Abstract

A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.

Original languageEnglish
JournalDynamical Systems
Volume33
Issue number1
Pages (from-to)136-158
ISSN1468-9367
DOIs
Publication statusPublished - 2017

Keywords

  • Singular perturbation theory
  • Canards
  • Iterative methods

Cite this

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title = "On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter",
abstract = "A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.",
keywords = "Singular perturbation theory, Canards, Iterative methods",
author = "Morten Br{\o}ns and Kristiansen, {Kristian Uldall}",
year = "2017",
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language = "English",
volume = "33",
pages = "136--158",
journal = "Dynamical Systems",
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On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter. / Brøns, Morten; Kristiansen, Kristian Uldall.

In: Dynamical Systems, Vol. 33, No. 1, 2017, p. 136-158.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter

AU - Brøns, Morten

AU - Kristiansen, Kristian Uldall

PY - 2017

Y1 - 2017

N2 - A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.

AB - A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.

KW - Singular perturbation theory

KW - Canards

KW - Iterative methods

U2 - 10.1080/14689367.2017.1313390

DO - 10.1080/14689367.2017.1313390

M3 - Journal article

VL - 33

SP - 136

EP - 158

JO - Dynamical Systems

JF - Dynamical Systems

SN - 1468-9367

IS - 1

ER -