Mathieu Desroches ; Jean-Pierre Françoise ; Lucile Mégret - Canard-induced loss of stability across a homoclinic bifurcation

arima:1989 - Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées, August 23, 2015, Volume 20 - 2015 - Special issue - Colloquium in Honor of Éric Benoît - https://doi.org/10.46298/arima.1989
Canard-induced loss of stability across a homoclinic bifurcation

Authors: Mathieu Desroches ; Jean-Pierre Françoise ; Lucile Mégret

his article deals with slow-fast systems and is, in some sense, a first approach to a general problem, namely to investigate the possibility of bifurcations which display a dramatic change in the phase portrait in a very small (on the order of 10−7 in the example presented here) change of a parameter. We provide evidence of existence of such a very rapid loss of stability on a specific example of a singular perturbation setting. This example is strongly inspired of the explosion of canard cycles first discovered and studied by E Benoît, J.-L. Callot, F. Diener and M. Diener. After some presentation of the integrable case to be perturbed, we present the numerical evidences for this rapid loss of stability using numerical continuation. We discuss then the possibility to estimate accurately the value of the parameter for which this bifurcation occurs.


Volume: Volume 20 - 2015 - Special issue - Colloquium in Honor of Éric Benoît
Published on: August 23, 2015
Submitted on: February 11, 2015
Keywords: Slow-fast systems, canard solutions, Lambert function, numerical continuation.,Systèmes lents-rapides, solutions canard, fonction de Lambert, continuation numérique,[INFO] Computer Science [cs],[MATH] Mathematics [math]


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