Victor Buchstaber (Steklov Mathematical Institute and the University of Manchester)
We discuss a non-linear ordinary differential equation, which is applicable to several problems in physics, mechanics and geometry. The best-known application of this equation is one of the so-called Josephson junction models. We use a double confluent Heun equation associated with this equation. We describe the families of polynomial and holomorphic solutions of this Heun equation. As a result, we obtain important special solutions of our equation. We describe explicitly the manifolds $M$ of these solutions. Our equation gives a dynamical system on a torus. We obtain formulas for the rotation numbers and the Poincaré maps of such dynamical systems with any parameters from $M$.