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| Invariant circles play an important role as barriers to transport in the dynamics of area-preserving maps. KAM theory guarantees the persistence of some circles for near-integrable maps, but far from the integrable case all circles can be destroyed. A standard method for determining the existence or nonexistence of a circle, Greene's residue criterion, requires the computation of long-period orbits, which can be difficult if the map has no reversing symmetry. We use a quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a Diophantine circle conjugate to rigid rotation, and the singularity of a norm of a derivative of the conjugacy to predict criticality. We study near-critical conjugacies for families of rotational invariant circles in generalizations of Chirikov's standard map.
A first goal is to obtain evidence to support the long-standing conjecture that when circles breakup they form cantori, as is known for twist maps by Aubry-Mather theory. A second goal is to support the conjecture that locally most robust circles have noble rotation numbers, even when the map is not reversible. Finally, we observe that the rotation number of the globally most robust circle generically appears to be a piecewise-constant function in two-parameter families of maps. |
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