Critical invariant circles in asymmetric and multiharmonic generalized standard maps

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.