An approximate renormalization-group transformation for Hamiltonian systems with three degrees of freedom

Abstract

We construct an approximate renormalization transformation that combines Kolmogorov-Arnold-Moser (KAM)and renormalization-group techniques, to analyze instabilities in Hamiltonian systems with three degrees of freedom. This scheme is implemented both for isoenergetically nondegenerate and for degenerate Hamiltonians. For the spiral mean frequency vector, we find numerically that the iterations of the transformation on nondegenerate Hamiltonians tend to degenerate ones on the critical surface. As a consequence, isoenergetically degenerate and nondegenerate Hamiltonians belong to the same universality class, and thus the corresponding critical invariant tori have the same type of scaling properties. We numerically investigate the structure of the attracting set on the critical surface and find that it is a strange nonchaotic attractor. We compute exponents that characterize its universality class.

10 pages typeset using REVTeX, 7 PS figures

References in corpus (6)

• A Renormalization Group for Hamiltonians: Numerical Results
• Kolmogorov-Arnold-Moser Renormalization-Group Approach to the Breakup of Invariant Tori in Hamiltonian Systems
• Kolmogorov-Arnold-Moser--Renormalization-Group Analysis of Stability in Hamiltonian Flows
• Universality for the breakup of invariant tori in Hamiltonian flows
• Critical Attractor and Universality in a Renormalization Scheme for Three Frequency Hamiltonian Systems
• A Version of Thirring's Approach to the KAM Theorem for Quadratic Hamiltonians with Degenerate Twist

Cited by in corpus (2)

• Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems
• Approximate renormalization for the break-up of invariant tori with three frequencies