ИСТИНА

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 (this system is called the Josephson junction): there could exist a supercurrent tunneling through this junction. We will discuss the reduction of the overdamped Josephson junction to a family of first order non-linear ordinary differential equations that defines a family of dynamical systems on two-torus. Physical problems of the Josephson junction led to studying the rotation number of the above-mentioned dynamical system on the torus as a function of the parameters and to the problem on the geometric description of the phase-lock areas: the level sets of the rotation number function ρ with non-empty interiors. Phase-lock areas were observed and studied for the first time by V.I.Arnold in the so-called Arnold family of circle diffeomorphisms at the beginning of 1970-ths. He has shown that in his family the phase-lock areas (which later became Arnold tongues) exist exactly for all the rational values of the rotation number. In our case the phase-lock areas exist only for integer rotation numbers (quantization effect). On their complement, which is an open set, the rotation number function ρ is an analytic submersion that induces its fibration by analytic curves. It appears that the family of dynamical systems on torus under consideration is equivalent to a family of second order linear complex differential equations on the Riemann sphere with two irregular singularities, the well-known double confluent Heun equations. This family of linear equations has the form LE=0, where L=L_{λ,μ,n} is a family of second order differential operators acting on germs of holomorphic functions of one complex variable. They depend on complex parameters λ, μ, n. The above-mentioned dynamical systems on torus correspond to the equations with real parameters satisfying the inequality λ+μ^2>0. The monodromy of the Heun equations is expressed in terms of the rotation number. We show that for all b,n∈C satisfying a certain “non-resonance condition” and for all parameter values λ,μ∈C, μ≠0 there exists an entire function f±:C→C (unique up to a constant factor) such that z^{−b}L(z^b f±(z^{±1}))=d_{0±}+d_{1±}z for some d_{0±},d_{1±}∈C. The constants d_{j,±} are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those values λ, μ, n and b for which the monodromy operator of the corresponding Heun equation has eigenvalue e^{2πib}. It also gives the description of those values λ, μ, n for which the monodromy is parabolic, i.e., has a multiple eigenvalue; they correspond exactly to the boundaries of the phase-lock areas. This implies the explicit description of the union of boundaries of the phase-lock areas as solutions of an explicit transcendental functional equation. For every θ∉Z we get a description of the set {ρ≡±θ(mod2Z)}. The talk will be accessible for a wide audience and devoted to different connections between physics, dynamical systems on two-torus and applications of analytic theory of complex linear differential equations.