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

V. M. Buchstaber ; A. A. Glutsyuk

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Order and chaos in dynamical systems, Tome 297 (2017), pp. 62-104.

Résumé

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 $\{\rho\equiv\pm\theta\pmod{2\mathbb Z}\}$.

Bibliographie

[1] Arnold V. I., Dopolnitelnye glavy teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978 | MR

[2] Arnold V. I., Ilyashenko Yu. S., “Obyknovennye differentsialnye uravneniya”, Dinamicheskie sistemy – 1, Itogi nauki i tekhniki. Sovr. probl. matematiki. Fund. napr., 1, VINITI, M., 1985, 7–140 | MR | Zbl

[3] Barone A., Paternò G., Physics and applications of the Josephson effect, J. Wiley and Sons, New York, 1982

[4] Buchstaber V. M., Glutsyuk A. A., “On determinants of modified Bessel functions and entire solutions of double confluent Heun equations”, Nonlinearity, 29:12 (2016), 3857–3870 | DOI | MR | Zbl

[5] Bukhshtaber V. M., Karpov O. V., Tertychnyi S. I., “Elektrodinamicheskie svoistva dzhozefsonovskogo perekhoda, obluchaemogo posledovatelnostyu $\delta $-impulsov”, ZhETF, 120:6 (2001), 1478–1485

[6] Bukhshtaber V. M., Karpov O. V., Tertychnyi S. I., “O svoistvakh differentsialnogo uravneniya, opisyvayuschego dinamiku silnoshuntirovannogo perekhoda Dzhozefsona”, UMN, 59:2 (2004), 187–188 | DOI | MR | Zbl

[7] Bukhshtaber V. M., Karpov O. V., Tertychnyi S. I., “Osobennosti dinamiki dzhozefsonovskogo perekhoda, smeschennogo sinusoidalnym SVCh-tokom”, Radioelektronika i elektrotekhnika, 51:6 (2006), 757–762

[8] Bukhshtaber V. M., Karpov O. V., Tertychnyi S. I., “Effekt kvantovaniya chisla vrascheniya”, TMF, 162:2 (2010), 254–265 | DOI | MR | Zbl

[9] Bukhshtaber V. M., Karpov O. V., Tertychnyi S. I., “Sistema na tore, modeliruyuschaya dinamiku perekhoda Dzhozefsona”, UMN, 67:1 (2012), 181–182 | DOI | MR | Zbl

[10] Bukhshtaber V. M., Tertychnyi S. I., “Semeistvo yavnykh reshenii uravneniya rezistivnoi modeli perekhoda Dzhozefsona”, TMF, 176:2 (2013), 163–188 | DOI | MR | Zbl

[11] Bukhshtaber V. M., Tertychnyi S. I., “Golomorfnye resheniya dvazhdy konflyuentnogo uravneniya Goina, assotsiirovannogo s RSJ-modelyu perekhoda Dzhozefsona”, TMF, 182:3 (2015), 373–404 | DOI | MR

[12] Bukhshtaber V. M., Tertychnyi S. I., “Zamechatelnaya posledovatelnost besselevykh matrits”, Mat. zametki, 98:5 (2015), 651–663 | DOI | MR | Zbl

[13] Bukhshtaber V. M., Tertychnyi S. I., “Avtomorfizmy prostranstva reshenii spetsialnykh dvazhdy konflyuentnykh uravnenii Goina”, Funkts. analiz i ego pril., 50:3 (2016), 12–33 | DOI | MR | Zbl

[14] Foote R. L., “Geometry of the Prytz planimeter”, Rep. Math. Phys., 42:1–2 (1998), 249–271 | DOI | MR | Zbl

[15] Foote R., Levi M., Tabachnikov S., “Tractrices, bicycle tire tracks, hatchet planimeters, and a 100-year-old conjecture”, Am. Math. Mon., 120:3 (2013), 199–216 | DOI | MR | Zbl

[16] Glutsyuk A. A., Kleptsyn V. A., Filimonov D. A., Schurov I. V., “O kvantovanii peremychek v uravnenii, modeliruyuschem effekt Dzhozefsona”, Funkts. analiz i ego pril., 48:4 (2014), 47–64 | DOI | MR | Zbl

[17] Ganning R., Rossi Kh., Analiticheskie funktsii mnogikh kompleksnykh peremennykh, Mir, M., 1969 | MR

[18] Ilin V. P., Kuznetsov Yu. I., Trekhdiagonalnye matritsy i ikh prilozheniya, Nauka, M., 1985 | MR

[19] Ilyashenko Yu. S., Lektsii Letnei shkoly “Dinamicheskie sistemy”, Poprad, Slovakiya, 2009

[20] Ilyashenko Yu. S., Ryzhov D. A., Filimonov D. A., “Zakhvat fazy dlya uravnenii, opisyvayuschikh rezistivnuyu model dzhozefsonovskogo perekhoda, i ikh vozmuschenii”, Funkts. analiz i ego pril., 45:3 (2011), 41–54 | DOI | MR | Zbl

[21] Josephson B. D., “Possible new effects in superconductive tunnelling”, Phys. Lett., 1:7 (1962), 251–253 | DOI | Zbl

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

[23] Likharev K. K., Ulrikh B. T., Sistemy s dzhozefsonovskimi kontaktami: Osnovy teorii, Izd-vo MGU, Moskva, 1978

[24] McCumber D. E., “Effect of ac impedance on dc voltage–current characteristics of superconductor weak-link junctions”, J. Appl. Phys., 39:7 (1968), 3113–3118 | DOI

[25] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (eds.), NIST handbook of mathematical functions, Cambridge Univ. Press, Cambridge, 2010 | MR | Zbl

[26] Shmidt V. V., Vvedenie v fiziku sverkhprovodnikov, 2-e izd., MTsNMO, M., 2000

[27] Shapiro S., Janus A. R., Holly S., “Effect of microwaves on Josephson currents in superconducting tunneling”, Rev. Mod. Phys., 36:1 (1964), 223–225 | DOI

[28] Slavyanov S., Lai V., Spetsialnye funktsii: Edinaya teoriya, osnovannaya na analize osobennostei, Nevskii dialekt, SPb., 2002

[29] Stewart W.C., “Current-voltage characteristics of Josephson junctions”, Appl. Phys. Lett., 12:8 (1968), 277–280 | DOI

[30] Tertychniy S. I., 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: Unlectured notes, E-print, 2005, arXiv: math-ph/0512058

[31] Tertychniy S. I., The modelling of a Josephson junction and Heun polynomials, E-print, 2006, arXiv: math-ph/0601064