ИСТИНА

In 1973 B. Josephson received Nobel Prize for discovering a new fundamental e ect in superconductivity concerning a sys- tem of two superconductors separated by a very narrow di-electric: 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 rst order di erential equations. Motivated by problems on Josephson e ect, 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 di erent 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 (identi ed with the average voltage through a long time in- terval). The 𝑟-th level set of the rotation number function is called the 𝑟-th phase-lock area, if it has non-empty inte- rior. Our family is very atypical: the phase-lock areas exist only for integer values of the rotation number, in di erence to the classical Arnold tongue picture (the quantization e ect discovered by V. M. Buchstaber, O. V. Karpov and S. I. Ter- tychnyi in 2010). The complexi cation of our family is the projectivization of a family of classical double con uent Heun equations: second order linear di erential equations on the Riemann sphere with two irregular singular points at 0 and in nity. 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 phase- lock areas and analytic conjectures on Heun equations and re- lations between them. We will present the state of art in this area including recent works of V. M. Buchstaber, S. I. Ter- tychnyi, A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, I. V. Schurov.