Critical Invariant Circles in Asymmetric and Multiharmonic Generalized Standard Maps

TL;DR

This paper investigates the persistence and destruction of invariant circles in generalized standard maps, using advanced numerical methods to analyze their robustness, criticality, and the formation of cantori, with implications for transport barriers in dynamical systems.

Contribution

It introduces a Fourier-based quasi-Newton scheme to numerically study invariant circles in non-reversible maps and explores their critical behavior and robustness properties.

Findings

Invariant circles tend to form cantori upon breakup.

Robustness of circles is inversely related to discriminant in quadratic fields.

Most robust circles often have noble rotation numbers.

Abstract

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.