We study a two-parameter family of nonautonomous ordinary differential equations on the 2-torus. This family models the Josephson effect in superconductivity. We study its rotation number as a function of the parameters and the Arnold tongues (also known as the phase locking domains) defined as the level sets of the rotation number that have nonempty interior. The Arnold tongues of this family of equations have a number of nontypical properties: they exist only for integer values of the rotation number, and the boundaries of the tongues are given by analytic curves. (These results were obtained by Buchstaber-Karpov-Tertychnyi and Ilyashenko-Ryzhov-Filimonov.) The tongue width is zero at the points of intersection of the boundary curves, which results in adjacency points. Numerical experiments and theoretical studies carried out by Buchstaber-Karpov-Tertychnyi and Klimenko-Romaskevich show that each Arnold tongue forms an infinite chain of adjacent domains separated by adjacency points and going to infinity in an asymptotically vertical direction. Recent numerical experiments have also shown that for each Arnold tongue all of its adjacency points lie on one and the same vertical line with integer abscissa equal to the corresponding rotation number. In the present paper, we prove this fact for an open set of two-parameter families of equations in question. In the general case, we prove a weaker claim: the abscissa of each adjacency point is an integer, has the same sign as the rotation number, and does not exceed the latter in absolute value. The proof is based on the representation of the differential equations in question as projectivizations of linear differential equations on the Riemann sphere and the classical theory of linear equations with complex time.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 48, No. 4, pp. 47–64, 2014
Original Russian Text Copyright © by A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, and I. V. Schurov
The present paper uses results obtained by I. V. Schurov under the support of the project no. 11-01-0239 “Invariant manifolds and asymptotic behavior of slow-fast mappings” within the program “The HSE scientific foundation” in 2012–2014. His studies were also supported in part by a grant from the Dynasty Foundation and by RFBR grant no. 12-01-31241-mol a. The research of A. A. Glutsyuk was supported in part by French grants ANR-08-JCJC-0130-01 and ANR-13-JS01-0010. The research of all the authors was supported in part by RFBR-CNRS joint grant no. 10-01-93115 NTsNIL a and by RFBR grants nos. 10-01-00739-a and 13-01-00969-a.
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Glutsyuk, A.A., Kleptsyn, V.A., Filimonov, D.A. et al. On the adjacency quantization in an equation modeling the Josephson effect. Funct Anal Its Appl 48, 272–285 (2014). https://doi.org/10.1007/s10688-014-0070-z
- Josephson effect in superconductivity
- ordinary differential equation on the torus
- rotation number
- Arnold tongue
- linear ordinary differential equation with complex time
- irregular singularity
- Stokes operator