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TMF, 2010, Volume 162, Number 2, Pages 254–265 (Mi tmf6468)  

This article is cited in 12 scientific papers (total in 13 papers)

Rotation number quantization effect

V. M. Buchstaberab, O. V. Karpovb, S. I. Tertychnyib

a Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
b National Research Institute for Physicotechnical and Radio Engineering Measurements, Moscow, Russia

Abstract: We study a class of dynamical systems on a torus that includes dynamical systems modeling the dynamics of the Josephson transition. For systems in this class, we introduce certain characteristics including a sequence of functions depending on the system parameters. We prove that if this sequence converges at a given point in the parameter space, then its limit is equal to the classical rotation number, and we then call this point a quantization point for the rotation number. We prove that the rotation number of such a system takes only integer values at a quantization point. Quantization areas are thus defined in the parameter space, and the problem of effectively describing them becomes an important part of characterizing the systems under study. We present graphs of the rotation number at quantization points and under conditions when it is not quantized (an example of a half-integer rotation number) and diagrams for quantization areas.

Keywords: dynamical system on a torus, rotation number, quantization, Josephson effect

DOI: https://doi.org/10.4213/tmf6468

Full text: PDF file (774 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2010, 162:2, 211–221

Bibliographic databases:

Document Type: Article
Received: 07.09.2009

Citation: V. M. Buchstaber, O. V. Karpov, S. I. Tertychnyi, “Rotation number quantization effect”, TMF, 162:2 (2010), 254–265; Theoret. and Math. Phys., 162:2 (2010), 211–221

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. Yu. S. Ilyashenko, D. A. Ryzhov, D. A. Filimonov, “Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations”, Funct. Anal. Appl., 45:3 (2011), 192–203  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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. A. M. Vershik, A. P. Veselov, A. A. Gaifullin, B. A. Dubrovin, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, D. V. Millionshchikov, S. P. Novikov, T. E. Panov, A. G. Sergeev, I. A. Taimanov, “Viktor Matveevich Buchstaber (on his 70th birthday)”, Russian Math. Surveys, 68:3 (2013), 581–590  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. 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
    6. A. Klimenko, O. Romaskevich, “Asymptotic properties of Arnold tongues and Josephson effect”, Mosc. Math. J., 14:2 (2014), 367–384  mathnet  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. 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
    11. 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
    12. Borisov A., Kilin A., Mamaev I., “Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard”, Int. J. Bifurcation Chaos, 29:3 (2019), 1930008  crossref  mathscinet  zmath  isi  scopus
    13. 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
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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