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On the adjacency quantization in the equation modelling the Josephson effect

We investigate two-parametric family of non-autonomous ordinary differential equations on the two-torus $$\dot x=\frac{dx}{dt}=ν\sin x + a + s \sin t, \ a,ν,s\in\rr; \ ν\neq0 \text {is fixed},$$ that model the Josephson effect from superconductivity. We study its rotation number as a function of parameters $(a,s)$ and its {\it Arnold tongues}: the level sets of the rotation number that have non-empty interior. Its Arnold tongues have many non-typical properties: they exist only for integer rotation numbers (V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi (2010); Yu.S.Ilyashenko, D.A.Ryzhov, D.A.Filimonov (2011)); their boundaries are given by pairs of analytic curves (V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi (2004, 2012)). Numerical experiments and theoretical investigations (V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi (2006); A.V.Klimenko and O.L.Romaskevich (2012)) 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 the adjacencies of each Arnold tongue have one and the same integer abscissa $a$ equal to the corresponding rotation number. We prove this fact for every fixed $ν$ with $|ν|\leq1$. 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 (V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi (2004); R.L.Foote (1998); Yu.S.Ilyashenko, D.A.Ryzhov, D.A.Filimonov (2011)), and the classical theory of complex linear equations.