An iterative method for the canard explosion in general planar systems

Morten Brøns

doi:10.3934/proc.2013.2013.77

Article Contents

2013, Volume 2013: 77-83. Doi: 10.3934/proc.2013.2013.77 open access

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An iterative method for the canard explosion in general planar systems

Morten Brøns^1,

1.
Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby

Received: August 31, 2012

Revised: March 31, 2013

Published: October 2013

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Abstract

The canard explosion is the change of amplitude and period of a limit cycle born in a Hopf bifurcation in a very narrow parameter interval. The phenomenon is well understood in singular perturbation problems where a small parameter controls the slow/fast dynamics. However, canard explosions are also observed in systems where no such parameter can obviously be identified. Here we show how the iterative method of Roussel and Fraser, devised to construct regular slow manifolds, can be used to determine a canard point in a general planar system of nonlinear ODEs. We demonstrate the method on the van der Pol equation, showing that the asymptotics of the method is correct, and on a templator model for a self-replicating system.

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Mathematics Subject Classification: Primary: 37G15, 93C70; Secondary: 34C05, 80A30.

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