Current browse context:
math.DS
Change to browse by:
References & Citations
Mathematics > Dynamical Systems
Title: Finding Birkhoff Averages via Adaptive Filtering
(Submitted on 27 Mar 2024)
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.
Submission history
From: Maximilian Ruth [view email][v1] Wed, 27 Mar 2024 20:53:15 GMT (27559kb,D)
Link back to: arXiv, form interface, contact.