Abstract:
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 $\rho$
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 $\rho$ 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 $\mathcal{L} E=0$, where $\mathcal{L}=\mathcal{L}_{\lambda,\mu,n}$
is a family of second order differential operators acting on germs of holomorphic functions of one complex variable. They depend on
complex parameters $\lambda$, $\mu$, $n$. The above-mentioned dynamical systems on torus correspond to the equations with
real parameters satisfying the inequality $\lambda+\mu^2>0$. The monodromy of the Heun equations is
expressed in terms of the rotation number. We show that
for all $b,n\in\mathbb{C}$ satisfying a certain “non-resonance condition” and for all parameter values $\lambda,\mu\in\mathbb{C}$, $\mu\neq0$
there exists an entire function $f_{\pm}:\mathbb{C}\to\mathbb{C}$ (unique up to a constant factor)
such that $z^{-b}\mathcal{L}(z^b f_{\pm}(z^{\pm1}))=d_{0\pm}+d_{1\pm}z$ for some
$d_{0\pm},d_{1\pm}\in\mathbb{C}$. The constants $d_{j,\pm}$ are expressed as functions of the parameters.
This result has several applications. First of all, it gives the description of those values $\lambda$, $\mu$, $n$ and $b$
for which the monodromy operator of the corresponding Heun equation has eigenvalue $e^{2\pi i b}$. It also gives the description
of those values $\lambda$, $\mu$, $n$ 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 $\theta\notin\mathbb{Z}$ we get a description of the set $\{\rho\equiv\pm\theta(mod2\mathbb{Z})\}$.
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.