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Topics in Representation Theory
Topics in Representation Theory
On dynamical systems on torus modeling Josephson junctionand Heun equations
On dynamical systems on torus modeling Josephson junctionand Heun equations
演讲者
Glutsyuk Alexey
时间
2026年04月17日 13:00 至 14:30
地点
A3-2a-302
线上
Zoom 242 742 6089
(BIMSA)
摘要
In 1962 B.Josephson (Nobel Prize 1973) predicted a tunnelling effectrelated to a system of two superconductors separated by a narrow dielectric: the so-called Josephson junction. This effect is the existence of asupercurrent crossing the junction and governed by equations discoveredby Josephson. The overdamped Josephson junction is modeled by a family of differential equations on two-dimensional torus that depends on three parameters: the abscissa B; the ordinate A and a fixed frequency ω of exterior forcing. The corresponding rotation number is a function of (B,A). The phase-lock areas are those its level sets that have non-empty interi-ors. They exist only for integer rotation numbers, as was discovered by V.M.Buchstaber, O.V.Karpov and S.I.Tertychnyi. Buchstaber and Terty-chnyi have shown that the model is equivalent to a family of second orderlinear differential equations: special double confluent Heun equations. Eachphase-lock area Lr with rotation number r ∈ Z is a garland of infinitelymany domains separated by points: white domains in the left figure. It wasshown by Yu.Bibilo and the speaker that those separation points that donot lie on the abscissa axis lie on the vertical line with abscissa rω. This was done by using relations to linear equations and isomonodromy deformations.
We present a survey of results and open questions.
We present a survey of results and open questions.