Application of Painlevé 3 equations to dynamical systems on 2-torus modeling Josephson junction

We consider a family of dynamical systems modeling overdamped Josephson junction in superconductivity. We focus on the family's rotation number as a function of parameters. Those level sets of the rotation number function that have non-empty interiors are called the phase-lock areas. It is known that each phase-lock area is an infinite garland of domains going to infinity in the vertical direction and separated by points. Those separation points that do not lie on the abscissa axis are called constrictions.
The model can be equivalently described via a family of linear systems (Josephson type systems) on the Riemann sphere. In the talk we will discuss isomonodromic deformations of the Josephson type systems – they are derived by Painlevé 3 equations. As an application of this approach, we present two new geometric results about the constrictions of the phase-lock areas solving two conjectures about them. We also present some open problems.
The talk is based on a joint work with A. A. Glutsyuk.