Kolmogorov-Arnold-Moser renormalization-group approach to the breakup of invariant tori in Hamiltonian systems

Abstract

We analyze the breakup of invariant tori in Hamiltonian systems with two degrees of freedom using a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. We consider a class of Hamiltonians quadratic in the action variables that is invariant under the chosen KAM transformations, following the approach of Thirring. The numerical implementation of the transformation shows that the KAM iteration converges up to the critical coupling at which the torus breaks up. By combining this iteration with a renormalization, consisting of a shift of resonances and rescalings of momentum and energy, we obtain a more efficient method that allows one to determine the critical coupling with high accuracy. This transformation is based on the physical mechanism of the breakup of invariant tori. We show that the critical surface of the transformation is the stable manifold of codimension one of a nontrivial fixed point, and we discuss its universality properties.