## 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 *L*_{r} = {*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 *∂**L*_{r} of the phase-lock area *L*_{r} with the line Λ_{r} = {*B* = *r**ω*} (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 area**L*_{r}*lie in its axis* Λ_{r}. 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 *L*_{r} with positive ordinates, the interval between them also lies in *L*_{r}. 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 *L*_{r} contains the infinite interval of the axis Λ_{r} issued upwards from the simple intersection in *∂**L*_{r} ∩Λ_{r} 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.

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## Notes

There is a misprint, missing 2

*π*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)].The transformation

*i**G*_{l}is equivalent to the right bemol transformation*b*_{e}from [16, the formula after (8)].

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## 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.

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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

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DOI: https://doi.org/10.1007/s10883-018-9411-1

### Keywords

- Josephson effect in superconductivity
- Ordinary differential equation on 2-torus
- Rotation number
- Phase-lock area
- Linear ordinary differential equation with complex time
- Irregular singularity
- Monodromy
- Stokes operator