  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.
