On phase-lock areas in a model of Josephson effect and double confluent Heun equations

In 1973 B. Josephson received Nobel Prize for discovering a new fundamental effect in superconductivity concerning a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction (called Josephson junction). One of the known models of the overdamped Josephson junction used a family of non-linear first order differential equations. Motivated by problems on Josephson effect, V. M. Buchstaber, O. V. Karpov and S. I. Tertychnyi studied this family in a series of papers (see JETP 2001 and later). They introduced and studied an equivalent family of dynamical systems on 2-torus. Motivated by dynamical systems point of view, Yu. S. Ilyashenko and J. Guckenheimer suggested and studied a subfamily of this family on 2-torus as an interesting slow-fast system in 2001. Analogous systems arose in different models of mathematics, mechanics and physics that a priori were not related one to the other.
The classical rotation number of the family of dynamical systems on 2-torus is a function of parameters of the family (identified with the average voltage through a long time interval). The $r$-th level set of the rotation number function is called the $r$-th phase-lock area, if it has non-empty interior. Our family is very atypical: the phase-lock areas exist only for integer values of the rotation number, in difference to the classical Arnold tongue picture (the quantization effect discovered by V. M. Buchstaber, O. V. Karpov and S. I. Tertychnyi in 2010). The complexification of our family is the projectivization of a family of classical double confluent Heun equations: second order linear differential equations on the Riemann sphere with two irregular singular points at 0 and infinity.
In the talk we will present results on geometric and analytic properties of the boundaries of the phase-lock areas. We will discuss several important geometric conjectures on the phaselock areas and analytic conjectures on Heun equations and relations between them. We will present the state of art in this area including recent works of V. M. Buchstaber, S. I. Tertychnyi, A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, I. V. Schurov.
Acknowledgements. Research of V. M. Buchstaber was supported by Russian Foundation for Basic Research (RFBR) grants 14-01-00506, 17-01-00192. Research of A. A. Glutsyuk was supported by Russian Science Foundation (RSF) grant 18- 41-05003, by Russian Foundation for Basic Research (RFBR) grants 13-01-00969-a, 16-01-00748, 16-01-00766, and by French National Research Agency (ANR) grant ANR-13-JS01-0010.