Hamiltonian-like phenomena in saddle-node bifurcations of invariant curves for plane diffeomorphisms. (English) Zbl 0561.58036

Chenciner, Alain

Singularities and dynamical systems, Proc. Int. Conf., Heraklion/Greece 1983, North-Holland Math. Stud. 103, 7-14 (1985).

[For the entire collection see Zbl 0547.00033.]
This paper is a brief summary of the author’s recent work on the dynamics of a generic two parameter family of maps near a degenerate Hopf bifurcation. Subject to some resonance conditions, the author concludes that the unfolding of such a map contains a large open set of parameter values for which the map ”looks like” a normal form. But there are other regions - in fact, infinitely many ”bubbles” in parameter space - for which the map has complicated dynamics associated to homoclinic tangles. In this sense, these regions give rise to zones of instability as occur in the dynamics of area preserving twist maps.

Reviewer: R.Devaney

MSC:

37G99

Local and nonlocal bifurcation theory for dynamical systems

Keywords:

Hopf bifurcation; unfolding; homoclinic tangles; area preserving twist maps

Citations:

Zbl 0547.00033