Josephson junction, Arnold tongues, and their adjacency points

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Victor Kleptsyn
CNRS, Rennes University
Tue, 03/13/2018 - 1:00pm - 2:00pm
RH 440R

The study of the equation on the 2-torus given by  
x’= sin x + a + b sin t
has been motivated by its relation to the Josephson junction in physics, as well as by purely mathematical reasons. For any values of the parameters a and b, one can consider the time-2\pi (period) map from the x-circle to itself, and study its properties, in particular, its rotation number.

Study of the Arnold tongues corresponding to this family, reveals a miracle: sometimes, their left and right boundaries intersect at a hourglass-type so-called adjacency point. Moreover, the a-coordinates of all these points turn out to be integers. My talk will be devoted to the geometry behind all of this, summarizing the works of many authors: Ilyashenko, Guckenheimer, Buchstaber, Karpov, Tertychnyi, Glutsyuk, Klimenko, Schurov, Filimonov, Romaskevich, Ryzhov, and myself.