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Thick Arnold tongues

Mark Levi, Alexey Okunev

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.

Thick Arnold tongues

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 . 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 . In the limiting case of zero inertia particles the dynamics is Hamiltonian with . For small but nonzero 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 , 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 , 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.

Paper Structure

This paper contains 23 sections, 33 theorems, 91 equations, 13 figures.

Table of Contents

  1. Introduction and Results
  2. Background
  3. Subject of this paper
  4. Main results
  5. Monotone families of circle maps with flat spots
  6. The Hamiltonian flow
  7. From particle dynamics to a circle map
  8. Reduction to a torus flow
  9. Poincaré map
  10. Construction of the family of circle maps.
  11. Proof of Theorem \ref{['t:main']}.
  12. Circle maps with flat spots
  13. Proof of monotonicity of the circle map family
  14. Incorporating external force in the fluid velocity field
  15. Properties of the Hamiltonian flow
  16. ...and 8 more sections

Key Result

Theorem 1.1

There exists $\gamma>0$ such that for any positive $\delta < \gamma$ there exists $\varepsilon_0 > 0$ such that, provided that $a$ and $b \in [\delta, \gamma]$ with $a \le 1.5b$ and $\varepsilon \in (0, \varepsilon_0)$, the following holds for e:MR with ${\bf v}$ given by e:fluid.

Figures (13)

  • Figure 1: A particle in a cellular flow
  • Figure 2: Two trajectories of (\ref{['e:MR']}) where ${\bf v}$ has a 3-fold symmetry. The trajectories are separated by stable manifolds of equilibrium points.
  • Figure 3: Trajectories of the system \ref{['e:MR']}, \ref{['e:fluid']} with $\sqrt { a ^2 + b ^2 } = 0.02$ and $\varepsilon = 1/25$.
  • Figure 4: Particle drift direction
  • Figure 5: Steepness of $m(\alpha)$ near the ends of plateaus.
  • ...and 8 more figures

Theorems & Definitions (72)

  • Theorem 1.1
  • Definition 1.2: Monotone family of circle maps with flat spots
  • Theorem 1.3
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Remark 3.4
  • proof : Proof of Theorem \ref{['t:main']}
  • Lemma 4.1
  • Lemma 4.2
  • ...and 62 more