Geodesic
Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 3, pp. 41-54

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In this work we study dynamical systems on the torus modeling Josephson junctions in the theory of superconductivity, and also perturbations of these systems. We show that, in the family of equations that describe resistively shunted Josephson junctions, phase lock occurs only for integer rotation numbers and propose a simple method for calculating the boundaries of the corresponding Arnold tongues. This part is a simplification of known results about the quantization of rotation number [4]. Moreover, we show that the quantization of rotation number only at integer points is a phenomenon of infinite codimension. Namely, there is an infinite set of independent perturbations of systems that give rise to countably many nondiscretely located phase-locking regions.
Keywords: differential equations on the torus, perturbation theory, Josephson effect, phase lock, quantization of rotation number, Arnold tongues. 
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     author = {Yu. S. Ilyashenko and D. A. Ryzhov and D. A. Filimonov},
     title = {Phase-lock effect for equations modeling resistively shunted {Josephson} junctions and for their perturbations},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {41--54},
     publisher = {mathdoc},
     volume = {45},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2011_45_3_a4/}
}
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Yu. S. Ilyashenko; D. A. Ryzhov; D. A. Filimonov. Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 3, pp. 41-54. http://geodesic.mathdoc.fr/item/FAA_2011_45_3_a4/