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Arnold Tongues in Area-Preserving Maps

Jing Zhou, Mark Levi

TL;DR

The paper investigates Arnold tongues in non-exact area-preserving cylinder maps with drift, showing that for a potential $V(x)=\delta x+\varepsilon F(x)$ the width of a $p/q$ tongue is bounded by $|\delta| \lesssim c\varepsilon^{\,[q/d]}$ when $F'$ is a trig polynomial of degree $d$. It develops a detailed remainder structure for iterates, proves the existence of $p/q$-periodic orbits via an implicit function framework, and derives a precise $\varepsilon$-expansion for the corresponding $y$-curve, with leading coefficients given by shift-averages of $f$; crucially, the leading term $\Delta_r(x)$ is a periodic trigonometric polynomial of degree at most $rd$, forcing $r>\left\lfloor q/d\right\rfloor$. This yields an explicit, sharp link between the harmonic content of the potential and resonance pinning, and connects to traveling waves in the discretized sine-Gordon equation. The results generalize Arnold–Keller–Levy phenomenology to area-preserving cylinder maps and provide a quantitative description of how higher harmonics stabilize or destabilize periodic orbits under drift.

Abstract

In the early 60's J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. 20 years later V. Arnold rediscovered a similar phenomenon on sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of objects where a similarly flavored behavior takes place: area-preserving maps of the cylinder. Speaking loosely, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to ``drift". The observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems.

Arnold Tongues in Area-Preserving Maps

TL;DR

The paper investigates Arnold tongues in non-exact area-preserving cylinder maps with drift, showing that for a potential the width of a tongue is bounded by when is a trig polynomial of degree . It develops a detailed remainder structure for iterates, proves the existence of -periodic orbits via an implicit function framework, and derives a precise -expansion for the corresponding -curve, with leading coefficients given by shift-averages of ; crucially, the leading term is a periodic trigonometric polynomial of degree at most , forcing . This yields an explicit, sharp link between the harmonic content of the potential and resonance pinning, and connects to traveling waves in the discretized sine-Gordon equation. The results generalize Arnold–Keller–Levy phenomenology to area-preserving cylinder maps and provide a quantitative description of how higher harmonics stabilize or destabilize periodic orbits under drift.

Abstract

In the early 60's J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. 20 years later V. Arnold rediscovered a similar phenomenon on sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of objects where a similarly flavored behavior takes place: area-preserving maps of the cylinder. Speaking loosely, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to ``drift". The observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems.
Paper Structure (7 sections, 4 theorems, 85 equations, 6 figures)

This paper contains 7 sections, 4 theorems, 85 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Results
  3. The Preliminaries
  4. Structure of the remainders
  5. The Existence of Periodic Orbits
  6. The Periodicity Lemma
  7. End of proof of the main theorem

Key Result

Theorem 1

Let $f(x)$ in (eq:cylindermap) be a trigonometric polynomial of degree $d$, and let $p\geq 0$, $q>0$ be integers. There exist positive constants $\overline{\varepsilon}$ and $c$ depending only on $q$ and $f$, such that for any $0<\varepsilon<\overline{\varepsilon}$, all $p/q$ periodic orbits of (eq:

Figures (6)

  • Figure 1: Non-exact area-preserving cylinder map with $\delta > 0$.
  • Figure 2: (A): the Frenkel-Kontorova model; (B): the tilt added, leading to the non-exact cylinder map; (C): tilt interpreted as torque acting on coupled pendula.
  • Figure 3: Discretized sine-Gordon equation: pendula with nearest-neighbor torsional coupling.
  • Figure 4: Josephson junctions: single and coupled. Voltage across the junction is proportional to $\langle \dot x \rangle$.
  • Figure 5: Left: Arnold tongue for the Birkhoff periodic point with the rotation number $\mu = p/q$ is exponentially narrow for trigonometric polynomials. Right: for Diophantine $\mu$ one has an invariant KAM curve only for the drift $\delta = 0$, i.e. the tongue has zero width.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Theorem 1: Width of Arnold tongues
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Remark 5
  • proof : Proof of Theorem \ref{['thm:ycurve']}
  • Lemma 2: Periodicity
  • ...and 1 more