Asymptotics of rotation number in a system describing Josephson function

A. Klimenko, jointly with O. Romaskevich, obtained a result concerning asymptotics of the Arnold tongues boundaries for a two-parametric family of vector fileds on a 2-torus. This family arises as a model describing effects emerging in a Josephson contact under oscullating electromagnetic field. Namely, it is shown that two boundaries of Arnold tongue corresponding to any integer rotation number are analytical curves that have inifinitely many intersections, and that these curves are asymptotically equivalent to graphs of Bessel functions appropriately scaled and shifted. The method developed for this problem is based on construction of Gronwall-type inequalities; it can be applicable to similar problems.

References

1. A. Klimenko, O. Romaskevich, “Asymptotic properties of Arnold tongues and Josephson effect”, Moscow Math. J., 14:2 (2014), 367–384, arXiv: 1305.6746