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On the adjacency quantization in the equation modelling the Josephson effect | Alexey Glutsyuk
; Dmitry Filimonov
; Victor Kleptsyn
; Ilya Schurov
; | Date: |
30 Jan 2013 | Abstract: | We investigate two-parametric family of non-autonomous ordinary differential
equations on the two-torus $$dot x=frac{dx}{dt}=
usin x + a + s sin t,
a,
u,sin
r;
u
eq0 ext {is fixed},$$ that model the Josephson effect
from superconductivity. We study its rotation number as a function of
parameters $(a,s)$ and its {it Arnold tongues}: the level sets of the rotation
number that have non-empty interior. Its Arnold tongues have many non-typical
properties: they exist only for integer rotation numbers (V.M.Buchstaber,
O.V.Karpov, S.I.Tertychnyi (2010); Yu.S.Ilyashenko, D.A.Ryzhov, D.A.Filimonov
(2011)); their boundaries are given by pairs of analytic curves
(V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi (2004, 2012)). Numerical
experiments and theoretical investigations (V.M.Buchstaber, O.V.Karpov,
S.I.Tertychnyi (2006); A.V.Klimenko and O.L.Romaskevich (2012)) show that each
Arnold tongue forms an infinite chain of adjacent domains separated by
adjacency points and going to infinity in asymptotically vertical direction.
Recent numerical experiments had also shown that the adjacencies of each Arnold
tongue have one and the same integer abscissa $a$ equal to the corresponding
rotation number. We prove this fact for every fixed $
u$ with $|
u|leq1$. In
the general case we prove a weaker statement: the abscissa of each adjacency
point is integer; it has the same sign, as the rotation number; its modulus is
no greater than that of the rotation number. The proof is based on the
representation of the differential equations under consideration as
projectivizations of complex linear differential equations on the Riemann
sphere (V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi (2004); R.L.Foote (1998);
Yu.S.Ilyashenko, D.A.Ryzhov, D.A.Filimonov (2011)), and the classical theory of
complex linear equations. | Source: | arXiv, 1301.7159 | Services: | Forum | Review | PDF | Favorites |
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