

Seminar on analytic theory of differential equations
May 3, 2017 14:30, Moscow, Steklov Mathematical Institute, Room 440 (8 Gubkina)






Arnold tongues in the model of the Josephson effect and holomorphic solutions of the Heun equation
A. A. Glutsyuk^{} 
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Abstract:
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 phaselock area in the Josephson effect) is a set of parameters having a nonempty interior such that the rotation number equals $n$ on it (such areas correspond to integer values of $n$ only). Each phaselock 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 phaselock 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).

