Birkhoff averages and rotational invariant circles for area-preserving maps

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 such high accuracy has not been obtained in 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 does 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.