Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps

TL;DR

This paper introduces a new method using weighted Birkhoff averages to identify invariant circles and chaotic regions in area-preserving maps, overcoming limitations of previous symmetry-dependent techniques.

Contribution

The authors develop a symmetry-independent approach based on weighted Birkhoff averages for analyzing rotational invariant circles in area-preserving maps.

Findings

Method successfully identifies invariant circles without symmetry assumptions.

Applicable to Chirikov's standard map and three other well-studied cases.

Confirms Greene's conjecture about noble rotation numbers and robustness.

Abstract

Rotational invariant circles of area-preserving maps are an important and well-studied example of KAM tori. John Greene conjectured that the locally most robust rotational circles have rotation numbers that are noble, i.e., have continued fractions with a tail of ones, and that, of these circles, the most robust has golden mean rotation number. The accurate numerical confirmation of these conjectures relies on the map having a time reversal symmetry, and these methods cannot be applied to more general maps. In this paper, we develop a method based on a weighted Birkhoff average for identifying chaotic orbits, island chains, and rotational invariant circles that do not rely on these symmetries. We use Chirikov's standard map as our test case, and also demonstrate that our methods apply to three other, well-studied cases.