Skip to main content

On the adjacency quantization in an equation modeling the Josephson effect


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.

This is a preview of subscription content, access via your institution.


  1. V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations [in Russian], Regular and Chaotic Dynamics, MCCME, Higher College of Mathematics, Independent University of Moscow, Moscow, 1999; English version: Springer-Verlag, New York-Berlin-Heidelberg, 1996.

    Google Scholar 

  2. V. I. Arnold and Yu. S. Ilyashenko, “Ordinary differential equations,” in: Dynamical Systems-1, Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Fundamental’nye Napravleniya, vol. 1, VINITI, Moscow, 1985, 7–140; English transl.: in: Dynamical Systems I, Encyclopaedia Math. Sci. (1988), 1–148.

    Google Scholar 

  3. V. M. Buchstaber, O. V. Karpov, and S. I. Tertychnyi, “A system on a torus modelling the dynamics of a Josephson junction,” Uspekhi Mat. Nauk, 67:1(403) (2012), 181–182; English transl.: Russian Math. Surveys, 67:1 (2012), 178–180.

    Article  Google Scholar 

  4. V. M. Buchstaber, O. V. Karpov, and S. I. Tertychnyi, “Rotation number quantization effect,” Teoret. Mat. Fiz., 162:2 (2010), 254–265; English transl.: Theor. Math. Phys., 162:2 (2010), 211–222.

    Article  Google Scholar 

  5. V.M. Buchstaber, O. V. Karpov, and S. I. Tertychnyi, “Peculiarities of dynamics of a Josephson junction shifted by a sinusoidal SHF current,” Radiotekhnika i Elektronika, 51:6 (2006), 757–762.

    Google Scholar 

  6. V. M. Buchstaber, O. V. Karpov, and S. I. Tertychnyi, “On properties of the differential equation describing the dynamics of an overdamped Josephson junction,” Uspekhi Mat. Nauk, 59:2 (2004), 187–188; English transl.: Russian Math. Surveys, 59:2 (2004), 377–378.

    Article  MathSciNet  Google Scholar 

  7. Yu. S. Ilyashenko, Lectures of the summer school “Dynamical systems”, Poprad (Slovakia), 2009.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Yu. S. Ilyashenko and A. G. Khovanskii, “Galois groups, Stokes operators, and a theorem of Ramis,” Funkts. Anal. Prilozhen., 24:4 (1990), 31–42; English transl.: Functional Anal. Appl., 24:4 (1990), 286–296.

    MathSciNet  Google Scholar 

  10. K. K. Likharev and B. T. Ulrikh, Systems with Josephson Junctions: Basic Theory [in Russian], Izdat. MGU, Moscow, 1978.

    Google Scholar 

  11. N. N. Luzin, “On the approximate integration method due to Academician S. A. Chaplygin,” Uspekhi Mat. Nauk, 6:6(46) (1951), 3–27.

    MATH  MathSciNet  Google Scholar 

  12. W. Balser, W. B. Jurkat, and D. A. Lutz, “Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations,” J. Math. Anal. Appl., 71:1 (1979), 48–94.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Barone and G. Paterno, Physics and Applications of the Josephson Effect, John Wiley and Sons, New York-Chichester-Brisbane-Toronto-Singapore, 1982.

    Book  Google Scholar 

  14. R. L. Foote, “Geometry of the Prytz Planimeter,” Rep. Math. Phys., 42:1–2 (1998), 249–271.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Guckenheimer and Yu. S. Ilyashenko, “The duck and the devil: canards on the staircase,” Moscow Math. J., 1:1 (2001), 27–47.

    MATH  MathSciNet  Google Scholar 

  16. S. Shapiro, A. Janus, and S. Holly, “Effect of microwaves on Josephson currents in superconducting tunneling,” Rev. Mod. Phys., 36 (1964), 223–225.

    Article  Google Scholar 

  17. W. B. Jurkat, D. A. Lutz, and A. Peyerimhoff, “Birkhoff invariants and effective calculations for meromorphic linear differential equations,” J. Math. Anal. Appl., 53:2 (1976), 438–470.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Klimenko and O. Romaskevich, “Asymptotic properties of Arnold tongues and Josephson effect,” Moscow Math. J., 14:2 (2014), 367–384.

    MATH  MathSciNet  Google Scholar 

  19. Y. Sibuya, “Stokes phenomena,” Bull. Amer. Math. Soc., 83:5 (1977), 1075–1077.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. A. Glutsyuk.

Additional information


Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 48, No. 4, pp. 47–64, 2014

Original Russian Text Copyright © by A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, and I. V. Schurov

The present paper uses results obtained by I. V. Schurov under the support of the project no. 11-01-0239 “Invariant manifolds and asymptotic behavior of slow-fast mappings” within the program “The HSE scientific foundation” in 2012–2014. His studies were also supported in part by a grant from the Dynasty Foundation and by RFBR grant no. 12-01-31241-mol a. The research of A. A. Glutsyuk was supported in part by French grants ANR-08-JCJC-0130-01 and ANR-13-JS01-0010. The research of all the authors was supported in part by RFBR-CNRS joint grant no. 10-01-93115 NTsNIL a and by RFBR grants nos. 10-01-00739-a and 13-01-00969-a.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Glutsyuk, A.A., Kleptsyn, V.A., Filimonov, D.A. et al. On the adjacency quantization in an equation modeling the Josephson effect. Funct Anal Its Appl 48, 272–285 (2014).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:

Key words

  • 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