# Josephson Effect and Fast-Slow Systems

O. V. Romaskevich,
V. A. Kleptsyn,
I. V. Schurov

January, 2013

### Abstract

In order to model the processes taking place in systems with Josephson
contacts, a differential equation on a torus with three parameters is used.
One of the parameters of the system can be considered small and the methods
of the fast—slow systems theory can be applied. The properties of the
phase—lock areas — the subsets in the parameter space, in which the changing
of a current doesn’t affect the voltage — are important in practical
applications. The phase-lock areas coincide with the Arnold tongues of a
Poincare map along the period. A description of the limit properties of
Arnold tongues is given. It is shown that the parameter space is split into
certain areas, where the tongues have different geometrical structures due
to fast—slow effects. An efficient algorithm for the calculation of tongue
borders is elaborated. The statement concerning the asymptotic approximation
of borders by Bessel functions is announced.

Publication

*Nanostructures. Mathematical Physics and Modelling*, **8** (1), 31–46