Thick Arnold tongues

TL;DR

This paper investigates the dynamics of inertial particles in a cellular flow, revealing complex drift behaviors, the emergence of Arnold tongues with full measure plateaus, and unexpected mathematical features related to rotation numbers and Cantor sets.

Contribution

It introduces a new model of particle flow exhibiting Arnold tongues with full measure plateaus, contrasting classical examples, and analyzes the intricate drift and rotation number phenomena.

Findings

Most trajectories drift to infinity with nonzero inertia.

Arnold tongues correspond to rational drift slopes with full measure.

The complement of all tongues has zero measure and Hausdorff dimension.

Abstract

We introduce and study a physically motivated problem that exhibits interesting and perhaps unexpected mathematical features. A cellular flow is a two-dimensional Hamiltonian flow of the Hamiltonian $H(x, y) = \cos(x) \cos(y)$. We study a simple model of the dynamics of an inertial particle carried by such a flow, subject to viscous drag and to an additional constant external force $(b, a)$. In the limiting case of zero inertia particles the dynamics is Hamiltonian with $H(x, y) = \cos(x) \cos(y) - ax + by$. For small but nonzero $a, \ b $ there appear ``channels" of trajectories that wind their way to infinity, of small relative measure, while most trajectories remain periodic. By contrast, for nonzero inertia, no matter how small, almost all particle trajectories drift to infinity. Moreover, the asymptotic direction of this drift no longer coincides with the direction of forcing, and rather becomes Cantor-like function of the forcing direction $a/b$, and with an unexpected feature: the plateaus of this function occupy a set of full measure. Moreover, the complement to this set has zero Hausdorff dimension. In a two-parameter representation (one parameter being the forcing direction $a/b$, the other the drag coefficient), this gives rise to Arnold tongues, the tongues corresponding to rational slopes of drift. However, unlike Arnold's example, the complement to the union of all tongues has zero measure. This is explained by the behavior of rotation number for monotone families of circle maps with flat spots.