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Funktsional. Anal. i Prilozhen., 2014, Volume 48, Issue 4, Pages 47–64 (Mi faa3161)  

This article is cited in 5 scientific papers (total in 5 papers)

On the Adjacency Quantization in an Equation Modeling the Josephson Effect

A. A. Glutsyukabc, V. A. Kleptsynd, D. A. Filimonovec, I. V. Shchurovc

a Independent University of Moscow
b CNRS — Unit of Mathematics, Pure and Applied
c National Research University "Higher School of Economics", Moscow
d Institute of Mathematical Research of Rennes
e Moscow Institute of Physics and Technology

Abstract: We study a two-parameter family of nonautonomous ordinary differential equations on the 2-torus. This family models the Josephson effect in superconductivity. We study its rotation number as a function of the parameters and the Arnold tongues (also known as the phase locking domains) defined as the level sets of the rotation number that have nonempty interior. The Arnold tongues of this family of equations have a number of nontypical properties: they exist only for integer values of the rotation number, and the boundaries of the tongues are given by analytic curves. (These results were obtained by Buchstaber–Karpov–Tertychnyi and Ilyashenko–Ryzhov–Filimonov.) The tongue width is zero at the points of intersection of the boundary curves, which results in adjacency points. Numerical experiments and theoretical studies carried out by Buchstaber–Karpov–Tertychnyi and Klimenko–Romaskevich show that each Arnold tongue forms an infinite chain of adjacent domains separated by adjacency points and going to infinity in an asymptotically vertical direction. Recent numerical experiments have also shown that for each Arnold tongue all of its adjacency points lie on one and the same vertical line with integer abscissa equal to the corresponding rotation number. In the present paper, we prove this fact for an open set of two-parameter families of equations in question. In the general case, we prove a weaker claim: the abscissa of each adjacency point is an integer, has the same sign as the rotation number, and does not exceed the latter in absolute value. The proof is based on the representation of the differential equations in question as projectivizations of linear differential equations on the Riemann sphere and the classical theory of linear equations with complex time.

Keywords: Josephson effect in superconductivity, ordinary differential equation on the torus, rotation number, Arnold tongue, linear ordinary differential equation with complex time, irregular singularity, monodromy, Stokes operator

Funding Agency Grant Number
National Research University Higher School of Economics 11-01-0239
Dynasty Foundation
Russian Foundation for Basic Research 12-01-31241-_
10-01-93115 _a
10-01-00739-
13-01-00969-
Agence Nationale de la Recherche ANR-08-JCJC-0130-01
ANR-13-JS01-0010


DOI: https://doi.org/10.4213/faa3161

Full text: PDF file (275 kB)
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English version:
Functional Analysis and Its Applications, 2014, 48:4, 272–285

Bibliographic databases:

Document Type: Article
UDC: 517.925.7
Received: 30.01.2013

Citation: A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, I. V. Shchurov, “On the Adjacency Quantization in an Equation Modeling the Josephson Effect”, Funktsional. Anal. i Prilozhen., 48:4 (2014), 47–64; Funct. Anal. Appl., 48:4 (2014), 272–285

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. M. Buchstaber, S. I. Tertychnyi, “Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction”, Theoret. and Math. Phys., 182:3 (2015), 329–355  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    2. Kleptsyn V., Okunev A., Schurov I., Zubov D., Katsnelson M.I., “Chiral Tunneling Through Generic One-Dimensional Potential Barriers in Bilayer Graphene”, Phys. Rev. B, 92:16 (2015), 165407  crossref  isi  elib  scopus
    3. Buchstaber V.M., Glutsyuk A.A., “On determinants of modified Bessel functions and entire solutions of double confluent Heun equations”, Nonlinearity, 29:12 (2016), 3857–3870  crossref  mathscinet  zmath  isi  elib  scopus
    4. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The HessAppelrot Case and Quantization of the Rotation Number”, Regul. Chaotic Dyn., 22:2 (2017), 180–196  mathnet  crossref  mathscinet
    5. V. M. Buchstaber, A. A. Glutsyuk, “On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect”, Proc. Steklov Inst. Math., 297 (2017), 50–89  mathnet  crossref  crossref  mathscinet  isi  elib
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