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**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.).**
*(French)*
Zbl 0712.58044

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 situation of the saddle-node invariant curve of the corresponding normal form.

This paper constitutes an important contribution to the knowledge of such two parameter families of diffeomorphisms, with in particular the description of the chain of infinitely many bubbles and its properties. However it would be more informative to complete the references by those giving more ancient results on one parameter families when a bubble is crossed. Such results are related to “exceptional” resonant bifurcation cases, deduced from Cigala normal forms. In particular, figures 20 were described in 1973, 1980 for rotation numbers 1/3, 1/4.

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 situation of the saddle-node invariant curve of the corresponding normal form.

This paper constitutes an important contribution to the knowledge of such two parameter families of diffeomorphisms, with in particular the description of the chain of infinitely many bubbles and its properties. However it would be more informative to complete the references by those giving more ancient results on one parameter families when a bubble is crossed. Such results are related to “exceptional” resonant bifurcation cases, deduced from Cigala normal forms. In particular, figures 20 were described in 1973, 1980 for rotation numbers 1/3, 1/4.

Reviewer: C.Mira

### MSC:

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

37G05 | Normal forms for dynamical systems |

57R50 | Differential topological aspects of diffeomorphisms |

70K40 | Forced motions for nonlinear problems in mechanics |

### Keywords:

resonance; map; iteration; invariant sets; rotation number; diffeomorphism; periodic orbit; bifurcation
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\textit{A. Chenciner}, Publ. Math., Inst. Hautes Étud. Sci. 66, 5--91 (1988; Zbl 0712.58044)

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