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Tr. Mat. Inst. Steklova, 2017, Volume 297, Pages 62–104 (Mi tm3794)  

On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect

V. M. Buchstaberab, A. A. Glutsyukcd

a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
b All-Russian Scientific Research Institute for Physical and Radio-Technical Measurements (VNIIFTRI), Mendeleevo, Solnechnogorskii raion, Moscow oblast, 141570 Russia
c CNRS (UMR 5669, UMPA, École normale supérieure de Lyon, 46, allée d'Italie, 69364 Lyon Cedex 07, France; Interdisciplinary Scientific Center J.-V. Poncelet, Bol'shoi Vlas'evskii per. 11, Moscow, 119002 Russia), France
d National Research University "Higher School of Economics", ul. Myasnitskaya 20, Moscow, 101000 Russia

Abstract: We study a family of double confluent Heun equations of the form $\mathcal LE=0$, where $\mathcal L=\mathcal L_{\lambda,\mu,n}$ is a family of second-order differential operators acting on germs of holomorphic functions of one complex variable. They depend on complex parameters $\lambda,\mu$, and $n$. The restriction of the family to real parameters satisfying the inequality $\lambda+\mu^2>0$ is a linearization of the family of nonlinear equations on the two-torus that model the Josephson effect in superconductivity. We show that for all $b,n\in\mathbb C$ satisfying a certain “non-resonance condition” and for all parameter values $\lambda,\mu\in\mathbb C$, $\mu\neq0$, there exists an entire function $f_\pm\colon\mathbb C\to\mathbb C$ (unique up to a constant factor) such that $z^{-b}\mathcal L(z^bf_\pm(z^{\pm1}))=d_{0\pm}+d_{1\pm}z$ for some $d_{0\pm},d_{1\pm}\in\mathbb C$. The constants $d_{j,\pm}$ are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those values $\lambda,\mu,n$, and $b$ for which the monodromy operator of the corresponding Heun equation has eigenvalue $e^{2\pi ib}$. It also gives the description of those values $\lambda,\mu$, and $n$ for which the monodromy is parabolic, i.e., has a multiple eigenvalue. We consider the rotation number $\rho $ of the dynamical system on the two-torus as a function of parameters restricted to a surface $\lambda+\mu^2=\mathrm{const}$. The phase-lock areas are its level sets with nonempty interior. For general families of dynamical systems, the problem of describing the boundaries of the phase-lock areas is known to be very complicated. In the present paper we include the results in this direction that were obtained by methods of complex variables. In our case the phase-lock areas exist only for integer rotation numbers (quantization effect), and their complement is an open set. On their complement the rotation number function is an analytic submersion that induces its fibration by analytic curves. The above-mentioned result on parabolic monodromy implies the explicit description of the union of boundaries of the phase-lock areas as solutions of an explicit transcendental functional equation. For every $\theta\notin\mathbb Z$ we get a description of the set $\{\rho\equiv\pm\theta\pmod{2\mathbb Z}\}$.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00506
13-01-00969-а
16-01-00748
16-01-00766
Agence Nationale de la Recherche ANR-13-JS01-0010
The first author was supported in part by the Russian Foundation for Basic Research (project no. 14-01-00506). The second author was supported in part by the Russian Foundation for Basic Research (project nos. 13-01-00969-a, 16-01-00748, and 16-01-00766) and by the French National Research Agency (project no. ANR-13-JS01-0010).


DOI: https://doi.org/10.1134/S0371968517020042

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English version:
Proceedings of the Steklov Institute of Mathematics, 2017, 297, 50–89

Bibliographic databases:

Document Type: Article
UDC: 517.925.7
Received: September 3, 2016

Citation: 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”, Order and chaos in dynamical systems, Collected papers. On the occasion of the 125th anniversary of the birth of Academician Dmitry Victorovich Anosov, Tr. Mat. Inst. Steklova, 297, MAIK Nauka/Interperiodica, Moscow, 2017, 62–104; Proc. Steklov Inst. Math., 297 (2017), 50–89

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