Quantitative quasiperiodicity

Abstract

The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, B N (f ) ∶= Σ N −1 n=0 f (x n ) N of a function f along a length N ergodic trajectory (x n ) of a function T converge to the space average ∫ f dµ, where µ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We use a modified average of f (x n ) by giving very small weights to the "end" terms when n is near 0 or N − 1. When (x n ) is a trajectory on a quasiperiodic torus and f and T are C ∞ , our Weighted Birkhoff average (denoted WB N (f )) converges "super" fast to ∫ f dµ with respect to the number of iterates N , i.e. with error decaying faster than N −m for every integer m. Our goal is to show that our Weighted Birkhoff average is a powerful computational tool, and this paper illustrates its use for several examples where the quasiperiodic set is one or two dimensional. In particular, we compute rotation numbers and conjugacies (i.e. changes of variables) and their Fourier series, often with 30-digit accuracy. d j=1 a j ρ j = 0, then every a j = 0. We then say such a ρ is irrational.Let T be a C ∞ quasiperiodic map. The quasiperiodicity persists for most small perturbations by the Kolmogorov-Arnold-Moser theory. We believe that quasiperiodicity is one of only three types of invariant sets with a dense trajectory that can occur in typical smooth maps. The other two types are periodic sets and chaotic sets. See [1] for the statement of our formal conjecture of this triumvirate. For example, quasiperiodicity occurs in a system of weakly coupled oscillators, in which there is an invariant smooth *

Simply improved averaging for coupled oscillators and weakly nonlinear waves

“…(22) is a good approximation because it was shown in [36] that, provided smoothness of f , the convergence speed is also faster than any power of N , unless the rotation vector of the quasiperiodicity (or equivalently, the frequencies ω 1 , · · · , ω d and ν 1 , · · · , ν n ) is from a measure-zero set 2 .…”

“…To perform the improved averaging (53), numerical averaging is employed for the same reason as discussed in Section 5.2. The only difference is, as the time averaging operator is no longer over a period but lim T →∞ 1 T T 0 , we use the method of weighted Birkhoff averaging ( [36], see also eq.22 and 23) instead of composite trapezoidal rule to approximate the time averages. While it is significantly slower when compared to the analytically averaged problem, the computation is still up-scaled because its total time remains bounded and independent of ǫ for simulation till O(ǫ −1 ) physical time.…”

Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem

“…In the series of papers [2,[17][18][19] we have developed techniques to characterize quasiperiodic orbits. We summarize them in this section used for our computation of this paper.…”

“…See [19] for a more general statement. See in particular [18] for details and a discussion of how the method relates to other approaches. Essentially the same weighting function as w [1] is discussed by Laskar [21] in the Remark 2 of the Annex (i.e.…”

Effective bounds for the measure of rotations

“…Due to the relevance of this topological invariant, during the last years, many numerical methods have been developed for this purpose. We refer for example to the works [7,41,45,46,51,56] and to [17] for a remarkable method with infinite order convergence, based on Birkhoff averages. As illustrated in Section 6, the KAM theorems presented in this paper allows us to obtain a rigorous enclosure (as tight as required) for the rotation number of a map, which is interesting in order to rigorously validate the numerical approximations performed with any of the mentioned numerical methods.…”