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

Abstract

We consider a class of Hamiltonians with three degrees of freedom that can be mapped into quasi-periodically driven pendulums. The purpose of this paper is to determine the threshold of the break-up of invariant tori with a specific frequency vector. We apply two techniques : the frequency map analysis and renormalizationgroup methods. The renormalization transformation acting on a Hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the Hamiltonian and a rescaling of phase space around the considered torus. We give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break-up of invariant tori. Furthermore, we obtain by these methods numerical values of the threshold of the break-up of the last invariant torus.

Stochastic ionization through noble tori: Renormalization results

“…The main conjecture of the renormalization approach is that if the torus exists for a given Hamiltonian H, the iterates R n H of the renormalization map acting on H converge to some integrable Hamiltonian H 0 . This conjecture is supported by analytical results in the perturbative regime [20,21], and by numerical results [15,14]. For a oneparameter family of Hamiltonians {H F }, the critical amplitude of the perturbation F c (ω) is determined by the following conditions :…”

Thresholds to chaos and ionization for the hydrogen atom in rotating fields

“…The critical thresholds are obtained by iterating the renormalization transformation R. The main conjecture of the renormalization approach is that if the torus exists for a given Hamiltonian H, the iterates R n H of the renormalization map acting on H converge to some integrable Hamiltonian H 0 . This conjecture is supported by analytical results in the perturbative regime [4,24], and by numerical results [5,18,20]. For a one-parameter family of Hamiltonians {H F }, the critical amplitude of the perturbation F c (ω) is determined by the following conditions :…”

“…[12] for the chaos threshold in the CP problem. We use this criterion in order to study the stability in the different regions of phase space as a function of the magnetic field ω c (for small values of the field ω c ) and the eccentricity of the initial orbit e. However, since this criterion is purely empirical, we use another method to validate or refine the results : we use the renormalization method which has proved to be a very powerful and accurate method for determining chaos thresholds in this type of models [5,18,19,20]. We compare the results given by both methods in the region where the criterion applies and we use the renormalization map to compute chaos thresholds when Eq.…”

Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom

“…For Hamiltonian systems with two and three degrees of freedom, renormalizationgroup transformations in the framework of Ref. [1] have been successfully applied for the determination of the threshold of the break-up of invariant tori [2,3,4]. They have also been used for the analysis of the properties of critical invariant tori (at the threshold of the break-up) [2,5,6,7,8].…”

“…For degenerate Hamiltonians with three degrees of freedom, a renormalization transformation has been defined for the spiral mean frequency vector. This transformation allows one to determine accurately the thresholds of break-up of tori by comparison with Laskar's Frequency Map Analysis [3].…”