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

  • Yu. S. Ilyashenko
  • D. A. Ryzhov
  • D. A. Filimonov

DOI: 10.1007/s10688-011-0023-8

Cite this article as:
Ilyashenko, Y.S., Ryzhov, D.A. & Filimonov, D.A. Funct Anal Its Appl (2011) 45: 192. doi:10.1007/s10688-011-0023-8


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.

Key words

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

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Yu. S. Ilyashenko
    • 1
    • 2
    • 3
  • D. A. Ryzhov
    • 4
  • D. A. Filimonov
    • 5
  1. 1.Cornell UniversityIthacaUSA
  2. 2.Moscow State UniversityMoscowRussia
  3. 3.Steklov Mathematical InstituteIndependent University of MoscowMoscowRussia
  4. 4.Chebyshev LaboratorySaint-Petersbourg State UniversitySaint-PetersbourgRussia
  5. 5.Moscow State University of Railway EngineeringMoscowRussia

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