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Funktsional. Anal. i Prilozhen., 2011, Volume 45, Issue 3, Pages 41–54 (Mi faa3045)  

This article is cited in 16 scientific papers (total in 16 papers)

Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations

Yu. S. Ilyashenkoabcd, D. A. Ryzhove, D. A. Filimonovf

a Moscow State University
b Independent University of Moscow
c Steklov Mathematical Institute
d Cornell University, USA
e Chebyshev Laboratory, Saint-Petersburg State University
f 5.Moscow State University of Railway Engineering

Abstract: In this work we study dynamical systems on the torus modeling Josephson junctions in the theory of superconductivity, and also perturbations of these systems. We show that, in the family of equations that describe resistively shunted Josephson junctions, phase lock occurs only for integer rotation numbers and propose a simple method for calculating the boundaries of the corresponding Arnold tongues. This part is a simplification of known results about the quantization of rotation number [4]. Moreover, we show that the quantization of rotation number only at integer points is a phenomenon of infinite codimension. Namely, there is an infinite set of independent perturbations of systems that give rise to countably many nondiscretely located phase-locking regions.

Keywords: differential equations on the torus, perturbation theory, Josephson effect, phase lock, quantization of rotation number, Arnold tongues.


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English version:
Functional Analysis and Its Applications, 2011, 45:3, 192–203

Bibliographic databases:

UDC: 517.923+517.925.54
Received: 03.12.2010

Citation: Yu. S. Ilyashenko, D. A. Ryzhov, D. A. Filimonov, “Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations”, Funktsional. Anal. i Prilozhen., 45:3 (2011), 41–54; Funct. Anal. Appl., 45:3 (2011), 192–203

Citation in format AMSBIB
\by Yu.~S.~Ilyashenko, D.~A.~Ryzhov, D.~A.~Filimonov
\paper Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations
\jour Funktsional. Anal. i Prilozhen.
\yr 2011
\vol 45
\issue 3
\pages 41--54
\jour Funct. Anal. Appl.
\yr 2011
\vol 45
\issue 3
\pages 192--203

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    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, O. V. Karpov, S. I. Tertychnyi, “A system on a torus modelling the dynamics of a Josephson junction”, Russian Math. Surveys, 67:1 (2012), 178–180  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. Engelbrecht J.R. Mirollo R., “Structure of long-term average frequencies for Kuramoto oscillator systems”, Phys. Rev. Lett., 109:3 (2012), 034103, 5 pp.  crossref  adsnasa  isi  elib  scopus
    4. V. M. Buchstaber, S. I. Tertychnyi, “Explicit solution family for the equation of the resistively shunted Josephson junction model”, Theoret. and Math. Phys., 176:2 (2013), 965–986  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Kleptsyn V.A., Romaskevich O.L., Schurov I.V., “Effekt Dzhozefsona i bystro-medlennye sistemy”, Nanostruktury. Matematicheskaya fizika i modelirovanie, 2013, no. 2, 31–46  elib
    6. A. Klimenko, O. Romaskevich, “Asymptotic properties of Arnold tongues and Josephson effect”, Mosc. Math. J., 14:2 (2014), 367–384  mathnet  crossref  mathscinet
    7. A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, I. V. Shchurov, “On the Adjacency Quantization in an Equation Modeling the Josephson Effect”, Funct. Anal. Appl., 48:4 (2014), 272–285  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. 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
    9. 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
    10. 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
    11. Glutsyuk A., Rybnikov L., “On Families of Differential Equations on Two-Torus With All Phase-Lock Areas”, Nonlinearity, 30:1 (2017), 61–72  crossref  mathscinet  zmath  isi  scopus
    12. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Sluchai Gessa–Appelrota i kvantovanie chisla vrascheniya”, Nelineinaya dinam., 13:3 (2017), 433–452  mathnet  crossref  mathscinet  elib
    13. 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  mathnet  crossref  mathscinet  zmath
    14. Xu C., Boccaletti S., Guan Sh., Zheng Zh., “Origin of Bellerophon States in Globally Coupled Phase Oscillators”, Phys. Rev. E, 98:5 (2018), 050202  crossref  mathscinet  isi
    15. Borisov A. Mamaev I., “Rigid Body Dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520  mathscinet  isi
    16. 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
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