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An iterative method for the canard explosion in general planar systems

Pages: 77 - 83, Issue special, November 2013

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Morten Brøns - Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark (email)

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.

Keywords:  Canards, singular perturbations, slow manifolds, templator.
Mathematics Subject Classification:  Primary: 37G15, 93C70; Secondary: 34C05, 80A30.

Received: September 2012;      Revised: April 2013;      Published: November 2013.

 References