Measuring quasiperiodicity

TL;DR

This paper introduces a modified numerical Birkhoff average that greatly accelerates convergence for quasiperiodic trajectories, aiding in distinguishing chaos from quasiperiodicity and computing rotation numbers in dynamical systems.

Contribution

The authors develop a weighted Birkhoff average method that speeds up convergence and introduce the Embedding Continuation Method for computing rotation numbers more simply.

Findings

Achieved a convergence speedup factor of 10^25 for 30-digit precision.

Effectively distinguishes quasiperiodic from chaotic trajectories.

Provides a new, simpler method for computing rotation numbers.

Abstract

The Birkhoff Ergodic Theorem asserts under mild conditions that Birkhoff averages (i.e. time averages computed along a trajectory) converge to the space average. For sufficiently smooth systems, our small modification of numerical Birkhoff averages significantly speeds the convergence rate for quasiperiodic trajectories -- by a factor of $10^{25}$ for 30-digit precision arithmetic, making it a useful computational tool for autonomous dynamical systems. Many dynamical systems and especially Hamiltonian systems are a complex mix of chaotic and quasiperiodic behaviors, and chaotic trajectories near quasiperiodic points can have long near-quasiperiodic transients. Our method can help determine which initial points are in a quasiperiodic set and which are chaotic. We use our {\bf weighted Birkhoff average} to study quasiperiodic systems, to distinguishing between chaos and quasiperiodicity, and for computing rotation numbers for self-intersecting curves in the plane. Furthermore we introduce the Embedding Continuation Method which is a significantly simpler, general method for computing rotation numbers.