Resonant elimination of a couple of invariant closed curves in the neighborhood of a degenerate Hopf bifurcation of diffeomorphisms of \({\mathbb{R}}^ 2\). (English) Zbl 0653.58027

Chenciner, A.

Dynamical systems, Proc. IIASA Workshop, Sopron/Hung. 1985, Lect. Notes Econ. Math. Syst. 287, 3-9 (1987).

[For the entire collection see Zbl 0632.00007.]
In some papers [see e.g. Singularities and dynamical systems, Proc. Int. Conf., Heraklion/Greece 1983, North-Holland Math. Stud. 103, 7-14 (1985; Zbl 0561.58036)] the author studied the dynamics in generic two-parameter families \({\mathcal P}=(P_{\mu,a})\) of local diffeomorphisms of \((R^ 2,0)\) (fixing 0) having properties which allow to show that they provide a good dissipative analogue of a generic area preserving local diffeomorphism F of \((R^ 2,0)\) having an elliptic fixed point at the origin 0. It appears that such a diffeomorphism can be treated as a perturbation of a certain family of diffeomorphisms leaving invariant each circle centered at 0 and which can be thought of as a one-parameter family of circle rotations degenerating at 0, the rotation number varying monotonically with the parameter. The results [see the author, loc. cit.] concern the relation of invariant sets of F (or \({\mathcal P})\) with so- called “good” irrational rotation numbers. In the reviewed paper the similar results for invariant sets whose rotation numbers belong to a sequence of “good” rationals are announced.

Reviewer: W.Kryszewski

MSC:

37G99	Local and nonlocal bifurcation theory for dynamical systems
37C80	Symmetries, equivariant dynamical systems (MSC2010)
57R50	Differential topological aspects of diffeomorphisms

Keywords:

Hopf bifurcation; dynamical systems; two-parameter families of local diffeomorphisms; invariant sets

Citations:

Zbl 0632.00007; Zbl 0561.58036