Bifurcations de points fixes elliptiques. I: Courbes invariantes. (French) Zbl 0566.58025

Chenciner, Alain

Publ. Math., Inst. Hautes Étud. Sci. 61, 67-127 (1985).

This is the first part of a work whose aim is to show that one-parameter families of diffeomorphisms of \({\mathbb{R}}^ 2\) undergoing a bifurcation where two invariant curves neatly cancel each other (as in the vector- field case) play, in the dissipative realm, the role of rotations of the circle in the K.A.M. theory of area-preserving plane diffeomorphisms.
In this paper, it is shown that in certain generic 2-parameter families of local plane diffeomorphisms unfolding a local diffeomorphism with an elliptic fixed point, there are always ”many” 1-parameter subfamilies undergoing such a neat bifurcation, in the same way as a generic area- preserving plane diffeomorphism always has, in the neighborhood of an elliptic fixed point, ”many” smooth invariant curves on which it is smoothly conjugated to a rotation. The proof uses normal forms and Russmann’s translated curve theorem in a version derived by Herman from Hamilton’s implicit function theorem.
The second part appeared in Invent. Math. 80, 81-106 (1985). The third part will appear shortly.

MSC:

37G99

Local and nonlocal bifurcation theory for dynamical systems

Keywords:

one-parameter families of diffeomorphisms; bifurcation; area-preserving plane diffeomorphisms

References:

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