Commun Nonlinear Sci Numer Simulat
Invariant circles play an important role as barriers to transport in the dynamics of area-pre- serving 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 numer- ically 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 cir- cles 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-integra- ble theory applies. A second goal is to support the conjecture that locally most robust cir- cles 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 qua- dratic 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.
Meiss, James and Fox, Adam Merritt, "Critical invariant circles in asymmetric and multiharmonic generalized standard maps" (2014). Applied Mathematics Faculty Contributions. 8.