Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Phase-locking in dynamical systems and quantum mechanics

Artem Alexandrov, Alexey Glutsyuk, Alexander Gorsky

Abstract

In this study, we discuss the Prufer transform that connects the dynamical system on the torus and the Hill equation, which is interpreted as either the equation of motion for the parametric oscillator or the Schrodinger equation with periodic potential. The structure of phase-locking domains in the dynamical system on torus is mapped into the band-gap structure of the Hill equation. For the parametric oscillator, we provide the relation between the non-adiabatic Hannay angle and the Poincare rotation number of the corresponding dynamical system. In terms of quantum mechanics, the integer rotation number is connected to the quantization number via the Milne quantization approach and exact WKB. Using recent results concerning the exact WKB approach in quantum mechanics, we discuss the possible non-perturbative effects in the dynamical systems on the torus and for parametric oscillator. The semiclassical WKB is interpreted in the framework of a slow-fast dynamical system. The link between the classification of the coadjoint Virasoro orbits and the Hill equation yields a classification of the phase-locking domains in the parameter space in terms of the classification of Virasoro orbits. Our picture is supported by numerical simulations for the model of the Josephson junction and Mathieu equation.

Phase-locking in dynamical systems and quantum mechanics

Abstract

In this study, we discuss the Prufer transform that connects the dynamical system on the torus and the Hill equation, which is interpreted as either the equation of motion for the parametric oscillator or the Schrodinger equation with periodic potential. The structure of phase-locking domains in the dynamical system on torus is mapped into the band-gap structure of the Hill equation. For the parametric oscillator, we provide the relation between the non-adiabatic Hannay angle and the Poincare rotation number of the corresponding dynamical system. In terms of quantum mechanics, the integer rotation number is connected to the quantization number via the Milne quantization approach and exact WKB. Using recent results concerning the exact WKB approach in quantum mechanics, we discuss the possible non-perturbative effects in the dynamical systems on the torus and for parametric oscillator. The semiclassical WKB is interpreted in the framework of a slow-fast dynamical system. The link between the classification of the coadjoint Virasoro orbits and the Hill equation yields a classification of the phase-locking domains in the parameter space in terms of the classification of Virasoro orbits. Our picture is supported by numerical simulations for the model of the Josephson junction and Mathieu equation.
Paper Structure (28 sections, 3 theorems, 163 equations, 7 figures, 4 tables)

This paper contains 28 sections, 3 theorems, 163 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Dynamical systems on torus
  3. Flows on the torus
  4. Hill equation as dynamical system
  5. Josephson junction as illustrative example
  6. Mathieu equation as simplest example
  7. Hannay angle and rotation number
  8. Definition of non-adiabatic Hannay angle
  9. Relation between Hannay angle and rotation number
  10. Hill operator and coadjoint Virasoro orbits
  11. Classification of coadjoint orbits
  12. Coadjoint orbits in RSJ model
  13. Comments on the flows in the space of orbits
  14. Rotation number quantization and exact WKB
  15. Exact WKB
  16. ...and 13 more sections

Key Result

Theorem 1

Consider a Hill equation of the form: where $g_1(\tau)$ and $g_2(\tau)$ are real-valued $2\pi$-periodic functions. Consider the corresponding dynamical system, on the torus $\mathbb T^2$. Then for every solution $u(\tau)$ of the Hill equation, the number $N(T)$ of its zeros on the interval $[0,T)$ is related to the rotation number of the dynamical system on torus eq:app-Hill-torus-dynsys as

Figures (7)

  • Figure 1: (a) Trajectory $\mathcal{C}$ on torus that start at point $A$ and after one period ends at point $B$, the dashed circle represents the Poincaré section, (b) Poincaré map $h$ that maps $A\in S^{1}$ to the point $B\in S^{1}$
  • Figure 2: Rotation number $\rho$ as the function of $B$ for fixed value of $A$ for the RSJ model.
  • Figure 3: (a) largest Lyapunov exponent chart for the RSJ model, (b) phase-locking domains (shaded by dark blue) in the RSJ model, (c) band structure of Hill equation corresponding to RSJ model (stability zones shaded by dark blue)
  • Figure 4: (a) rotation number quantization for Mathieu system ($A=2$, $\omega=1$) (b) band structure of Mathieu equation (stability zones shaded by dark blue), (c) phase-locking domains (shaded by dark blue) in the Mathieu system
  • Figure 5: Phase-locking domains (shaded) for the Mathieu equation with small frequencies $\omega$. The line corresponds to $A=-B$ and separate two domains
  • ...and 2 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • proof