Funktsional'nyi Analiz i ego Prilozheniya
General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Funktsional. Anal. i Prilozhen.:

Personal entry:
Save password
Forgotten password?

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

This article is cited in 12 scientific papers (total in 12 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
Agence Nationale de la Recherche ANR-08-JCJC-0130-01


Full text: PDF file (275 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2014, 48:4, 272–285

Bibliographic databases:

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
\by A.~A.~Glutsyuk, V.~A.~Kleptsyn, D.~A.~Filimonov, I.~V.~Shchurov
\paper On the Adjacency Quantization in an Equation Modeling the Josephson Effect
\jour Funktsional. Anal. i Prilozhen.
\yr 2014
\vol 48
\issue 4
\pages 47--64
\jour Funct. Anal. Appl.
\yr 2014
\vol 48
\issue 4
\pages 272--285

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Glutsyuk A.A. Netay I.V., “On Spectral Curves and Complexified Boundaries of the Phase-Lock Areas in a Model of Josephson Junction”, J. Dyn. Control Syst.  crossref  mathscinet  isi
    2. 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
    3. 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
    4. 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
    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
    6. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Sluchai GessaAppelrota i kvantovanie chisla vrascheniya”, Nelineinaya dinam., 13:3 (2017), 433–452  mathnet  crossref  mathscinet  elib
    7. 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  zmath
    8. A. Borisov, I. Mamaev, “Rigid body dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520  mathscinet  isi
    9. A. A. Glutsyuk, “On constrictions of phase-lock areas in model of overdamped Josephson effect and transition matrix of the double-confluent Heun equation”, J. Dyn. Control Syst., 25:3 (2019), 323–349  crossref  isi
    10. A. V. Malyutin, “The Rotation Number Integer Quantization Effect in Braid Groups”, Proc. Steklov Inst. Math., 305 (2019), 182–194  mathnet  crossref  crossref  mathscinet  isi  elib
    11. S. I. Tertychnyi, “Solution space monodromy of a special double confluent Heun equation and its applications”, Theoret. and Math. Phys., 201:1 (2019), 1426–1441  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. Yu. P. Bibilo, A. A. Glutsyuk, “On families of constrictions in the model of an overdamped Josephson junction”, Russian Math. Surveys, 76:2 (2021), 360–362  mathnet  crossref  crossref  isi  elib
  •     Functional Analysis and Its Applications
    Number of views:
    This page:417
    Full text:171
    First page:53

    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2022