Strange attractor for the renormalization flow for invariant tori of Hamiltonian systems with two generic frequencies

Abstract

We analyze the stability of invariant tori for Hamiltonian systems with two degrees of freedom by constructing a transformation that combines Kolmogorov-Arnold-Moser theory and renormalization-group techniques. This transformation is based on the continued fraction expansion of the frequency of the torus. We apply this transformation numerically for arbitrary frequencies that contain bounded entries in the continued fraction expansion. We give a global picture of renormalization flow for the stability of invariant tori, and we show that the properties of critical (and near critical) tori can be obtained by analyzing renormalization dynamics around a single hyperbolic strange attractor. We compute the fractal diagram, i.e., the critical coupling as a function of the frequencies, associated with a given one-parameter family.

10 pages, 5 figures

References in corpus (5)

• 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
• A Version of Thirring's Approach to the KAM Theorem for Quadratic Hamiltonians with Degenerate Twist

Cited by in corpus (5)

• Thresholds to Chaos and Ionization for the Hydrogen Atom in Rotating Fields
• Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems
• Scaling law for the critical function of an approximate renormalization
• Application of renormalization to the dynamics of a particle in a infinite square-well potential driven by an external field
• Stochastic ionization through noble tori: Renormalization results