Canards et enlacements

Benoît, Éric

doi:10.1007/bf02699131

Cited by 69 publications

(139 citation statements)

References 4 publications

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“…The dynamics of slow-fast systems (1.1) near generic folded equilibria was first studied by Benoit [4]. He proved the existence of canards, trajectories that continue along the unstable sheet of the critical manifold after approaching the folded equilibrium, in the cases that the equilibrium is a saddle or node.…”

Section: Singularities Folds and Jumpsmentioning

confidence: 99%

“…4) as a case study to investigate the bifurcations of relaxation oscillations [15,16,17]. Although this example does not contain all of the different types of generic bifurcations relaxation oscillations can undergo, it has led to a better understanding of how local degeneracies in the slow-fast decomposition of these periodic orbits play a role in the global bifurcation structure.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcations of relaxation oscillations

Guckenheimer

2004

Normal Forms, Bifurcations and Finiteness Problems in Differential Equations

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We are still far from a comprehensive theory of bifurcation in dynamical systems with multiple time scales. However, systematic application of geometric methods and study of examples have produced descriptions of varied phenomena. These lectures present a selective summary of some of what has been discovered, concentrating on periodic orbits called relaxation oscillations. We focus upon key aspects of the phenomena, avoiding mathematical details and making the analysis as simple as possible. The dynamics of reduced systems plays a central role in our discussion.

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Section: Singularities Folds and Jumpsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bifurcations of relaxation oscillations

Guckenheimer

2004

Normal Forms, Bifurcations and Finiteness Problems in Differential Equations

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“…Les méthodes géométriques non standard (voir [3], [4], [5]) ont donné sur ce type de systèmes un certain nombre de résultats généraux qui montraient l'importance des canards dans ces problèmes. Cependant, ils n'ont pas permis de répondreà une question fondamentale : l'existence de canards au voisinage d'un point pseudo-singulier de type noeud.…”

Section: Introductionunclassified

“…-Dans ce paragraphe, on reprend les définitions et notations de [3], [4], [5]. Dans ces anciens articles, le paramètre εétait considéré comme fixé, infinitésimal (au sens de l'analyse non standard).…”

Section: Introductionunclassified