Periodiccanard trajectories with multiple segments following the unstablepart of critical manifold

Alexander M. Krasnosel'skii; Edward O'Grady; Alexei Pokrovskii; Dmitrii I. Rachinskii

doi:10.3934/dcdsb.2013.18.467

Article Contents

2013, Volume 18, Issue 2: 467-482. Doi: 10.3934/dcdsb.2013.18.467

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Periodic canard trajectories with multiple segments following the unstable part of critical manifold

1.
Institute for Information Transmission Problems, 19 Bolshoi Karetny, Moscow 127994, Russian Federation, National Research University Higher School of Economics, 20 Myasnitskaya Street, Moscow 101000
2.
Department of Applied Mathematics, University College Cork
3.
Department of Applied Mathematics, University College, Cork

Received: September 30, 2011

Revised: February 29, 2012

Published: February 2013

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Abstract

We consider a scalar fast differential equation which is periodically driven by a slowly varying input. Assuming that the equation depends on $n$ scalar parameters, we present simple sufficient conditions for the existence of a periodic canard solution, which, within a period, makes $n$ fast transitions between the stable branch and the unstable branch of the folded critical curve. The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has $n$ portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the length of the time intervals of the slow motion separated by the short time intervals of fast transitions between the branches are controlled by the parameters.

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Mathematics Subject Classification: Primary: 34E17, 34C25; Secondary: 34C55.

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