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Moscow Mathematical Journal

Volume 14, Issue 2, April–June 2014  pp. 367–384.

Asymptotic Properties of Arnold Tongues and Josephson Effect

Authors:  A. Klimenko (1) and O. Romaskevich (2)
Author institution:(1) Steklov Mathematical Institute of RAS and
National Research University Higher School of Economics
(2) National Research University Higher School of Economics
École Normale Supérieure de Lyon


A three-parametrical family of ODEs on a torus arises from a model of Josephson effect in a resistive case when a Josephson junction is biased by a sinusoidal microwave current. We study asymptotics of Arnold tongues of this family on the parametric plane (the third parameter is fixed) and prove that the boundaries of the tongues are asymptotically close to Bessel functions.

2010 Mathematics Subject Classification. 37E45, 34E05.

Keywords: Arnold tongues, Josephson effect, dynamical system on the torus, rotation number.

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