Q748988
type: Item
instance of: scholarly article
title: Bifurcations de points fixes elliptiques. III: Orbites périodiques de ``petites'' périodes et élimination résonnante des couples de courbes invariantes. (Bifurcations of elliptic fixed points. III: Periodic orbits with ``small'' periods and resonant elimination of pairs of invariant curves.)
zbMATH Open document ID: 0712.58044
DOI: 10.1007/BF02698927
author: Alain Chenciner
published in: Publications Mathématiques
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104028
review text: The paper is the third part of a general study devoted to generic two parameter families of smooth local diffeomorphism of \({\mathbb{R}}^ 2\), written as perturbations of normal forms. [For part I see ibid. 61, 67- 127 (1985; Zbl 0566.58025), for part II see Invent. Math. 80, 81-106 (1985; Zbl 0578.58031)]. The first paper was concerned by invariant sets, the rotation number of which is a ``good'' irrational number, in the sense that the diffeomorphism has an invariant curve, close to a circle on which the induced map is smoothly conjugate to the rotation. In a neighbourhood of the rotation center, it has the same singularities as those of the normal form. This paper deals with the case of invariant sets with ``good'' rational rotation number, defined equivalently in its first section. Every periodic orbit with a good rotation number is well ordered. In the second section, from the considered generic situation, these periodic orbits are interpreted as the ``souvenir of nearly resonances'', being obtained by bifurcation from resonant case. Some restriction of the diffeomorphism to a ring appears as a perturbation of a resonant normal form invariant by the simple rotation. An interpolation by a two parameters family of differential equations is considered, the study of which is transferred in an appendix. The bifurcation diagram in the parameter plane makes appear infinitely many ``bubbles'' arranged in a string along a curve related to the merging of two invariant curves for the family of normal forms. In the complementary part of these bubbles the diffeomorphism behaves like a normal form. The third and fourth sections show that the bubbles become very narrow near points corresponding to good rational rotation numbers. The last section is devoted to complex phenomena occurring for some parameter values belonging to resonant tongues. The diffeomorphism has an ordered hyperbolic periodic orbit with homoclinic points, not lying on an invariant curve, in an equivalent …
reviewed by: Christian Mira
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37G05
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 70K40
zbMATH DE Number: 4172008
zbMATH Keywords: resonance
zbMATH Keywords: map
zbMATH Keywords: iteration
zbMATH Keywords: invariant sets
zbMATH Keywords: rotation number
zbMATH Keywords: diffeomorphism
zbMATH Keywords: periodic orbit
zbMATH Keywords: bifurcation
MaRDI profile type: Publication
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OpenAlex ID: W1580206188
label: Bifurcations de points fixes elliptiques. III: Orbites périodiques de ``petites'' périodes et élimination résonnante des couples de courbes invariantes. (Bifurcations of elliptic fixed points. III: Periodic orbits with ``small'' periods and resonant elimination of pairs of invariant curves.)
description: scientific article; zbMATH DE number 4172008