  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 de la Llave's 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. The location of the largest gaps is compared to the maxima
of the potential when anti-integrable theory applies. A second goal is to
support the conjecture that locally most robust circles have noble rotation
numbers, even when the map is not reversible. We show that relative robustness
varies inversely with the discriminant for rotation numbers in quadratic
algebraic fields. 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.
