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Working paper

On the adjacency quantization in the equation modelling the Josephson effect

arxiv.org. math. Cornell University, 2013. No. 1301.7159.
Glutsyuk A., Filimonov D., Kleptsyn V., Schurov I.
In the present paper we investigate two-parametric family of nonautonomous ordinary differential equations on the two-torus that model the Josephson effect from superconductivity. We study its rotation number as a function of parameters and its Arnold tongues (also called phase locking domains): the level sets of the rotation number that have non-empty interior. The Arnold tongues of the equation under consideration have many non-typical properties: the phase locking happens only for integer values of the rotation number; the boundaries of the tongues are given by analytic curves, the tongues have zero width at the intersection points of the latter curves (this yield the adjacency points). Numerical experiments and theoretical investigations show that each Arnold tongue forms an infinite chain of adjacent domains separated by adjacency points and going to infinity in asymptotically vertical direction. Recent numerical experiments had also shown that for each Arnold tongue all its adjacency points lie on one and the same vertical line with the integer abscissa equal to the corresponding rotation number. In the present paper we prove this fact for some open domain of the two-parametric families of equations under consideration. In the general case we prove a weaker statement: the abscissa of each adjacency point is integer; it has the same sign, as the rotation number; its modulus is no greater than that of the rotation number. The proof is based on the representation of the differential equations under consideration as projectivizations of complex linear differential equations on the Riemann sphere, see, and the classical theory of complex linear equations.