On the Adjacency Quantization in an Equation Modeling the Josephson Effect

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.