Colloquim Mathematics - Prof. Henk Broer, University of Groningen

Title: KAM-theory from the dynamical systems point of view

Abstract:

Kolmogorov-Arnold-Moser theory on quasi-periodic invariant tori started around 1950 as one of the greatest contributions to classical mechanics of the 20th century. Later it turned out that multi- or quasi-periodicity occurs a lot in the general theory of dynamical systems, particularly in systems depending on parameters. In the colloquium this development will be sketched, in particular the contributions from the Groningen school of dynamical systems.

One of the themes concerns repeated quasi-periodic Hopf bifurcations, beginning with the one named after Hopf-Neimark-Sacker. In natural philosophy this sequence of bifurcations relates to the Hopf-Landau-Lifschitz-Ruelle-Takens scenario for the onset of turbulence in fluid mechanics. A central role is reserved for the Cantor set, a topological gadget that in the present setting has positive measure.