Kolmogorov-Arnold-Moser–Renormalization-Group Analysis of Stability in Hamiltonian Flows

1997/11/17 by M. Govin, C. Chandré, C. Chandre +1 · 1 citation
Computer Science · Physics and Astronomy · #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #Theoretical and Computational Physics #chao-dyn #nlin.CD

paper · pdf · doi:10.1103/physrevlett.79.3881

published as Phys. Rev. Lett. 79, 3881 (1997) · 4 pages, 2 figures, RevTex

openalex publication_date 1997/11/17 · arxiv created 1998/02/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the stability and breakup of invariant tori in Hamiltonian flows using a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. We implement the scheme numerically for a family of Hamiltonians quadratic in the actions to analyze the strong coupling regime. We show that the KAM iteration converges up to the critical coupling at which the torus breaks up. Adding a renormalization consisting of a rescaling of phase space and a shift of resonances allows us to determine the critical coupling with higher accuracy. We determine a nontrivial fixed point and its universality properties.

Citations

• The renormalization group: Critical phenomena and the Kondo problem
• PROOF OF A THEOREM OF A. N. KOLMOGOROV ON THE INVARIANCE OF QUASI-PERIODIC MOTIONS UNDER SMALL PERTURBATIONS OF THE HAMILTONIAN

Cited by

• Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems