Arnold tongues in the model of the Josephson effect and holomorphic solutions of the Heun equation

We consider a family of dynamical systems on the torus modeling the Josephson effect in superconductivity. The Arnold tongue of level $n$ (the $n$-th phase-lock area in the Josephson effect) is a set of parameters having a non-empty interior such that the rotation number equals $n$ on it (such areas correspond to integer values of $n$ only). Each phase-lock area is an infinite chain of domains separated by adjencies and going to the infinity in the vertical direction. Area's boundaries have a Bessel asymptotics. The family under consideration on the torus is equivalent to a family of double confluent Heun equations on the Riemann sphere having only two singular points, which are irregular.
In the talk we give a survey of open problems and results concerning the geometry of phase-lock areas obtained by complex methods. In particular, we are interested in the description of coordinates of adjencies (a recent short proof of the Buchstaber–Tertychniy conjecture on a partial description of ordinates of adjencies was obtained in our joint paper with Buchstaber and uses ideas of the hyperbolic theory of dynamical systems).