Finding Birkhoff Averages via Adaptive Filtering

TL;DR

The paper addresses the slow convergence of ergodic Birkhoff averages used to classify trajectories in Hamiltonian and symplectic maps. It introduces Birkhoff Reduced Residual Extrapolation (Birkhoff RRE), an adaptive filter-learning approach that casts the problem as a constrained least-squares task to learn optimal weights from a single trajectory, achieving faster convergence than the traditional weighted Birkhoff average. The learned filter not only accelerates classification but also enables extraction of rotation numbers and island counts via the roots of the filter polynomial, allowing Fourier parameterizations of invariant circles and islands. Demonstrations on the standard map and a stellarator field-line map show significant speedups and robust parameterization, suggesting practical impact for efficient analysis of high-dimensional Hamiltonian systems. The method relies on a conjecture about RRE residual behavior, with potential extensions to higher dimensions and proofs as natural avenues for future work.

Abstract

In many applications, one is interested in classifying trajectories of Hamiltonian systems as invariant tori, islands, or chaos. The convergence rate of ergodic Birkhoff averages can be used to categorize these regions, but many iterations of the return map are needed to implement this directly. Recently, it has been shown that a weighted Birkhoff average can be used to accelerate the convergence, resulting in a useful method for categorizing trajectories. In this paper, we show how a modified version the reduced rank extrapolation method (named Birkhoff RRE) can also be used to find optimal weights for the weighted average with a single linear least-squares solve.Using these, we classify trajectories with fewer iterations of the map than the standard weighted Birkhoff average. Furthermore, for the islands and invariant circles, a subsequent eigenvalue problem gives the number of islands and the rotation number. Using these numbers, we find Fourier parameterizations of invariant circles and islands. We show examples of Birkhoff RRE on the standard map and on magnetic field line dynamics.

Figures (6)

• Figure 1: Phase portrait of the standard map \ref{['eq:standard-map']}. Parameterizations of the invariant circles and islands are obtained via the methodology in Sec. \ref{['sec:method']}, using Algorithm \ref{['alg:adaptiveRRE']} with $\epsilon=0$, $\gamma=3$, $\delta=10^{-10}$, $K_{\mathrm{init}}=50$, $K_{\max}=600$, and $\Delta K = 50$. Invariant circles and islands are colored according to their rotation numbers, while trajectories classified as chaotic are plotted in black.
• Figure 2: (top) A test signal $\bm{h}(\theta) = e^{\cos 2\pi \theta}$, sampled at the points $\omega t$ where $\omega = (\sqrt 5 - 1)/2$. (middle) A discrete Fourier transform of the signal, showing peaks near the expected frequencies. (bottom) Three candidate $\bar{K}=11$ filters: the all-ones filter, a weighted Birkhoff filter, and an tuned filter to the first five roots of the frequency. We see that the all-ones filter is largest where there is a large amount of power if the signal, the weighted filter is small far from the zero frequency, and the tuned filter is zero exactly at the relevant frequencies. The absolute errors of the Birkhoff average for each filter are: (all-ones) $7.11 \times 10^{-2}$, (weighted Birkhoff) $7.38 \times 10^{-3}$, and (tuned) $2.72 \times 10^{-5}$.
• Figure 3: (a) A Poincaré plot of the standard map with $k=0.7$, colored by trajectory classification of integrable, chaotic, or indeterminate. (b) Convergence of the weighted Birkhoff doubling error with respect to trajectory length. Trajectories are colored by the ending point on this plot, with $R > 10^{-5}$ chaotic, $R < 10^{-11}$ integrable, and the rest indeterminate. (c) Convergence of RRE residual with respect to trajectory length. (d) The same as data as (b), on the same domain as (c). From (c) and (d), we see that the RRE residual appears to converge much more rapidly than the weighted Birkhoff average.
• Figure 4: Number of integrable trajectories of Fig. \ref{['fig:residual-convergence']} misclassified, with the tolerance of both WBA and RRE set to $10^{-11}$. We see that RRE converges to a low misclassification rate much more efficiently than WBA in the number of map iterations.
• Figure 5: (a) A Poincaré plot of the standard map, with three invariant circle with initial $x$ points $0.1$, $0.35$, and $0.4$. (b-d) Convergence plots for the absolute error of the learned frequencies from the polynomial filter vs filter length $K$. The absolute error is computed by comparing to the frequency learned from a $K=1500$ simulation. In (b), we see very rapid convergence for the smooth central circle. In (c), we see slower, but still rapid convergence of the learned frequencies. In (d), for an invariant circle near the edge of chaos, we see significantly slower convergence. The errors are approximately straight in this plot, indicating nealry exponential convergence of the roots.

Theorems & Definitions (13)

• Theorem 2.2: Das and YorkeDas2018 Thm 3.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Conjecture 3.4
• Theorem C.1
• Corollary C.2
• proof