# On Constrictions of Phase-Lock Areas in Model of Overdamped Josephson Effect and Transition Matrix of the Double-Confluent Heun Equation

• Published:

## Abstract

In 1973, B. Josephson received a Nobel Prize for discovering a new fundamental effect concerning a Josephson junction,—a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction. We will discuss the model of the overdamped Josephson junction, which is given by a family of first-order nonlinear ordinary differential equations on two-torus depending on three parameters: a fixed parameter ω (the frequency); a pair of variable parameters (B, A) that are called respectively the abscissa, and the ordinate. It is important to study the rotation number of the system as a function ρ = ρ(B, A) and to describe the phase-lock areas: its level sets Lr = {ρ = r} with non-empty interiors. They were studied by V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi, who observed in their joint paper in 2010 that the phase-lock areas exist only for integer values of the rotation number. It is known that each phase-lock area is a garland of infinitely many bounded domains going to infinity in the vertical direction; each two subsequent domains are separated by one point, which is called constriction (provided that it does not lie in the abscissa axis). Those points of intersection of the boundary Lr of the phase-lock area Lr with the line Λr = {B = rω} (which is called its axis) that are not constrictions are called simple intersections. It is known that our family of dynamical systems is related to appropriate family of double–confluent Heun equations with the same parameters via Buchtaber–Tertychnyi construction. Simple intersections correspond to some of those parameter values for which the corresponding “conjugate” double-confluent Heun equation has a polynomial solution (follows from results of a joint paper of V.M. Buchstaber and S.I.Tertychnyi and a joint paper of V.M. Buchstaber and the author). There is a conjecture stating that all the constrictions of every phase-lock areaLrlie in its axis Λr. This conjecture was studied and partially proved in a joint paper of the author with V.A.Kleptsyn, D.A.Filimonov, and I.V.Schurov. Another conjecture states that for any two subsequent constrictions in Lr with positive ordinates, the interval between them also lies in Lr. In this paper, we present new results partially confirming both conjectures. The main result states that for every $$r \in \mathbb {Z}\setminus \{0\}$$ the phase-lock area Lr contains the infinite interval of the axis Λr issued upwards from the simple intersection in Lr ∩Λr with the biggest possible ordinate. The proof is done by studying the complexification of the system under question, which is the projectivization of a family of systems of second-order linear equations with two irregular non-resonant singular points at zero and at infinity. We obtain new results on the transition matrix between appropriate canonical solution bases of the linear system; on its behavior as a function of parameters. A key result, which implies the main result of the paper, states that the off-diagonal terms of the transition matrix are both non-zero at each constriction. We also show that their ratio is real at the constrictions. We reduce the above conjectures on constrictions to the conjecture on negativity of the ratio of the latter off-diagonal terms at each constriction.

This is a preview of subscription content, log in via an institution to check access.

## Subscribe and save

Springer+ Basic
\$34.99 /Month
• Get 10 units per month
• 1 Unit = 1 Article or 1 Chapter
• Cancel anytime

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

## Notes

1. There is a misprint, missing 2π in the denominator, in analogous formulas in previous papers of the author with co-authors: [20, formula (2.2)], [7, the formula after (1.16)].

2. The transformation iGl is equivalent to the right bemol transformation be from [16, the formula after (8)].

## References

1. Arnold VI. Geometrical methods in the theory of ordinary differential equations, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 250. New York: Springer; 1988.

2. Arnold VI, Ilyashenko YuS. Ordinary differential equations. Dynamical Systems I, Encyclopaedia Math. Sci; 1988. p. 1–148.

3. Balser W, Jurkat WB, Lutz DA. Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations. J Math Anal Appl 1979;71(1):48–94.

4. Barone A, Paterno G. Physics and applications of the Josephson effect. New York: Wiley; 1982.

5. Bibilo Yu. Josephson effect and isomonodromic deformations. Preprint arXiv:1805.11759.

6. Buchstaber VM, Glutsyuk AA. On determinants of modified Bessel functions and entire solutions of double confluent Heun equations. Nonlinearity 2016;29:3857–70.

7. Buchstaber VM, Glutsyuk AA. On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect. Proc Steklov Inst Math 2017;297:50–89.

8. Buchstaber VM, Karpov OV, Tertychniy SI. Electrodynamic properties of a Josephson junction biased with a sequence of δ-function pulses. J Exper Theoret Phys 2001;93(6):1280–7.

9. Buchstaber VM, Karpov OV, Tertychnyi SI. On properties of the differential equation describing the dynamics of an overdamped Josephson junction. Russian Math Surveys 2004;59:2:377–8.

10. Buchstaber VM, Karpov OV, Tertychnyi SI. Peculiarities of dynamics of a Josephson junction shifted by a sinusoidal SHF current (in Russian). Radiotekhnika i Elektronika 2006;51:6:757–62.

11. Buchstaber VM, Karpov OV, Tertychnyi SI. The rotation number quantization effect. Theoret Math Phys 2010;162(2):211–21.

12. Buchstaber VM, Karpov OV, Tertychnyi SI. The system on torus modeling the dynamics of Josephson junction. Russ Math Surveys 2012;67(1):178–80.

13. Buchstaber VM, Tertychnyi SI. Explicit solution family for the equation of the resistively shunted Josephson junction model. Theoret and Math Phys 2013;176(2):965–86.

14. Buchstaber VM, Tertychnyi SI. Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction. Theoret Math Phys 2015;182:3:329–55.

15. Buchstaber VM, Tertychnyi SI. A remarkable sequence of Bessel matrices. Mathematical Notes 2015;98(5):714–24.

16. Buchstaber VM, Tertychnyi SI. Automorphisms of solution space of special double-confluent Heun equations. Funct Anal Appl 2016;50:3:176–92.

17. Buchstaber VM, Tertychnyi SI. Representations of the Klein group determined by quadruples of polynomials associated with the double confluent Heun equation. Math Notes 2018;103:3:357–71.

18. Foote RL. Geometry of the Prytz planimeter. Reports on Math Phys 1998;42:1/2: 249–71.

19. Foote RL, Levi M, Tabachnikov S. Tractrices, bicycle tire tracks, hatchet planimeters, and a 100-year-old conjecture. Amer Math Monthly 2013;103:199–216.

20. Glutsyuk AA, Kleptsyn VA, Filimonov DA, Schurov IV. On the adjacency quantization in an equation modeling the Josephson effect. Funct Analysis and Appl 2014;48(4):272–85.

21. Ilyashenko YuS. Lectures of the summer school “Dynamical systems”. Slovak Republic: Poprad; 2009.

22. Ilyashenko YuS, Filimonov DA, Ryzhov DA. Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations. Funct Analysis and its Appl 2011;45(3):192–203.

23. Ilyashenko YuS, Khovanskii AG. Galois groups, Stokes operators, and a theorem of Ramis. Functional Anal Appl 1990;24:4:286–96.

24. Josephson BD. Possible new effects in superconductive tunnelling. Phys Lett 1962; 1(7):251–3.

25. Jurkat WB, Lutz DA, Peyerimhoff A. Birkhoff invariants and effective calculations for meromorphic linear differential equations. J Math Anal Appl 1976;53(2): 438–70.

26. Klimenko A, Romaskevich OL. Asymptotic properties of Arnold tongues and Josephson effect. Mosc Math J 2014;14:2:367–84.

27. Likharev KK, Ulrikh BT. Systems with Josephson junctions: basic theory. Moscow: MGU; 1978.

28. McCumber DE. Effect of ac Impedance on dc voltage-current characteristics of superconductor weak-link junctions. J Appl Phys 1968;39(7):3113–8.

29. Schmidt VV. Introduction to physics of superconductors (in Russian). Moscow: MCCME; 2000.

30. Shapiro S, Janus A, Holly S. Effect of microwaves on Josephson currents in superconducting tunneling. Rev Mod Phys 1964;36:223–5.

31. Sibuya Y. Stokes phenomena. Bull Amer Math Soc 1977;83:1075–7.

32. Slavyanov SYu, Lay W. Special functions: a unified theory based on singularities. Oxford: Oxford University Press; 2000.

33. Stewart WC. Current-voltage characteristics of Josephson junctions. Appl Phys Lett 1968;12(8):277–80.

34. Tertychnyi SI. Long-term behavior of solutions of the equation $$\dot {\phi }+\sin \phi = f$$ with periodic f and the modeling of dynamics of overdamped Josephson junctions, Preprint arXiv:math-ph/0512058.

35. Tertychnyi SI. The modelling of a Josephson junction and Heun polynomials, Preprint arXiv:math-ph/0601064.

## Acknowledgements

I am grateful to V.M.Buchstaber for attracting my attention to problems on model of Josephson effect and helpful discussions. I am grateful to Yu.P.Bibilo for helpful discussions.

## Author information

Authors

### Corresponding author

Correspondence to A. A. Glutsyuk.

Supported by RSF grant 18-41-05003.

## Rights and permissions

Reprints and permissions

Glutsyuk, A.A. On Constrictions of Phase-Lock Areas in Model of Overdamped Josephson Effect and Transition Matrix of the Double-Confluent Heun Equation. J Dyn Control Syst 25, 323–349 (2019). https://doi.org/10.1007/s10883-018-9411-1