Teoreticheskaya i Matematicheskaya Fizika
TMF:
TMF, 2013,	Volume 176,	Number 2,	Pages 163–188	(Mi tmf8512)
Explicit solution family for the equation of the resistively shunted Josephson junction model
V. M. Buchstaberab, S. I. Tertychnyia
a All-Russian Scientific Research Institute of Physical-Technical and Radiotechnical Measurements, Mendeleevo, Moscow Oblast, Russia
b Steklov Mathematical Institute, RAS, Moscow, Russia
Abstract: We obtain and study a family of solutions of the equation $\dot\phi+\sin\phi =B+A\cos\omega t$, which is applicable to several problems in physics, mechanics, and geometry. We use polynomial solutions of double confluent Heun equations associated with this equation to construct the family. We describe the manifold $M_{\mathrm P}$ of parameters $(A,B,\omega)$ of these solutions and obtain explicit formulas for the rotation number and Poincaré map of the dynamical system on a torus corresponding to this equation with parameters $(A,B,\omega)\in M_{\mathrm rP}$.
Keywords: dynamical system on a torus, double confluent Heun equations, polynomial solution, rotation number, Poincaré map
DOI: https://doi.org/10.4213/tmf8512
Theoretical and Mathematical Physics, 2013, 176:2, 965–986
Citation: V. M. Buchstaber, S. I. Tertychnyi, "Explicit solution family for the equation of the resistively shunted Josephson junction model", TMF, 176:2 (2013), 163–188; Theoret. and Math. Phys., 176:2 (2013), 965–986
\Bibitem{BucTer13}
\by V.~M.~Buchstaber, S.~I.~Tertychnyi
\paper Explicit solution family for the~equation of the~resistively shunted Josephson junction model
\jour TMF
\mathnet{http://mi.mathnet.ru/tmf8512}
\crossref{https://doi.org/10.4213/tmf8512}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2013TMP...176..965B}
\jour Theoret. and Math. Phys.
\crossref{https://doi.org/10.1007/s11232-013-0085-2}
http://mi.mathnet.ru/eng/tmf8512
https://doi.org/10.4213/tmf8512
http://mi.mathnet.ru/eng/tmf/v176/i2/p163
Glutsyuk A.A. Netay I.V., "On Spectral Curves and Complexified Boundaries of the Phase-Lock Areas in a Model of Josephson Junction", J. Dyn. Control Syst.
A. Klimenko, O. Romaskevich, "Asymptotic properties of Arnold tongues and Josephson effect", Mosc. Math. J., 14:2 (2014), 367–384
V. M. Buchstaber, S. I. Tertychnyi, "Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction", Theoret. and Math. Phys., 182:3 (2015), 329–355
V. M. Buchstaber, S. I. Tertychnyi, "On a Remarkable Sequence of Bessel Matrices", Math. Notes, 98:5 (2015), 714–724
V. M. Buchstaber, S. I. Tertychnyi, "Automorphisms of the solution spaces of special double-confluent Heun equations", Funct. Anal. Appl., 50:3 (2016), 176–192
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
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", Proc. Steklov Inst. Math., 297 (2017), 50–89
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., 25:3 (2019), 323–349
A. V. Malyutin, "The Rotation Number Integer Quantization Effect in Braid Groups", Proc. Steklov Inst. Math., 305 (2019), 182–194
S. I. Tertychnyi, "Solution space monodromy of a special double confluent Heun equation and its applications", Theoret. and Math. Phys., 201:1 (2019), 1426–1441
Tertychniy S.I., "Symmetries of the Space of Solutions to Special Double Confluent Heun Equations of Integer Order", J. Math. Phys., 60:10 (2019), 103501
A. E. Artisevich, A. B. Shabat, "Tri teoremy o matritsakh Vandermonda", Vladikavk. matem. zhurn., 22:1 (2020), 5–12
V. M. Buchstaber, S. I. Tertychnyi, "Categories of Symmetry Groups of the Space of Solutions of the Special Doubly Confluent Heun Equation", Math. Notes, 110:5 (2021), 643–654
First page: 37