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Article

О квантовании перемычек в уравнении, моделирующем эффект Джозефсона

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.