Stochastic ionization through noble tori: Renormalization results

Abstract

We find that chaos in the stochastic ionization problem develops through the break-up of a sequence of noble tori. In addition to being very accurate, our method of choice, the renormalization map, is ideally suited for analyzing properties at criticality. Our computations of chaos thresholds agree closely with the widely used empirical Chirikov criterion.

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

“…However, this feature varies with the parameter e as it has been observed in Ref. [19]. For e ∈ [0, 0.8] and for ω c = 0, the regions near m:1 with large m become very stable (and this stabilization is increased by the field for low values of ω c ) and the diffusion of the trajectories is very limited.…”

“…However, this feature may vary with the parameter e as it has been observed in Ref. [19] and in Sec. III A for the negative twist region.…”

“…2 of Ref. [19]). The main feature of this frequency is that it remains noble as the parameters e and ω c are varied, in the sense that the tail of the continued fraction expansion of this frequency is a sequence of 1, or equivalently this frequency expresses like (aγ + b)/(cγ + d) where a, b, c, d are integers such that ad − bc = ±1.…”

“…[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

“…Remark : For the forced pendulum model (19), the Tribonacci torus is the most robust between the three invariant tori investigated in this paper, and the spiral mean torus is more robust than the τ -mean torus. However, this feature depends on the perturbation in a way that is not understood.…”