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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

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

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-ìîë_à10-01-93115 ÍÖÍÈË_a10-01-00739-à13-01-00969-à Agence Nationale de la Recherche ANR-08-JCJC-0130-01ANR-13-JS01-0010

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

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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

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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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
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
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
4. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hess–Appelrot Case and Quantization of the Rotation Number”, Regul. Chaotic Dyn., 22:2 (2017), 180–196
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
6. Borisov A. Mamaev I., “Rigid Body Dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520
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