Hans-Rudolf JAUSLIN

Laboratoire de Physique 
Universite de Bourgogne 
B.P. 400 F- 21011 Dijon 
France 
jauslin@u-bourgogne.fr

KAM-Renormalization Group approach to instabilities in Hamiltonian flows

Abstract : We present some recent work on the stability of KAM tori in Hamiltonian flows. The analysis is based on a combination of KAM transformations and renormalizati on group ideas. The scheme is implemented numerically for a family of Hamiltonians quadratic in the actions. We obtained results that indicate that the KAM iterati on can converge up to the critical coupling at which the torus breaks up. Adding a renormalization scheme consisting of a rescaling and a shift of the relevant resonances, we obtain a much more efficient method that allows to determine the critical coupling with high accuracy. We determine a non-trivial fixed point of the renormalization transformation, and discuss the associated universality properties.