200,809 research outputs found
Breathers in the weakly coupled topological discrete sine-Gordon system
Existence of breather (spatially localized, time periodic, oscillatory)
solutions of the topological discrete sine-Gordon (TDSG) system, in the regime
of weak coupling, is proved. The novelty of this result is that, unlike the
systems previously considered in studies of discrete breathers, the TDSG system
does not decouple into independent oscillator units in the weak coupling limit.
The results of a systematic numerical study of these breathers are presented,
including breather initial profiles and a portrait of their domain of existence
in the frequency-coupling parameter space. It is found that the breathers are
uniformly qualitatively different from those found in conventional spatially
discrete systems.Comment: 19 pages, 4 figures. Section 4 (numerical analysis) completely
rewritte
Linear difference equations, frieze patterns and combinatorial Gale transform
We study the space of linear difference equations with periodic coefficients
and (anti)periodic solutions. We show that this space is isomorphic to the
space of tame frieze patterns and closely related to the moduli space of
configurations of points in the projective space. We define the notion of
combinatorial Gale transform which is a duality between periodic difference
equations of different orders. We describe periodic rational maps generalizing
the classical Gauss map
Dynamics of a rational system of difference equations in the plane
We consider a rational system of first order difference equations in the
plane with four parameters such that all fractions have a common denominator.
We study, for the different values of the parameters, the global and local
properties of the system. In particular, we discuss the boundedness and the
asymptotic behavior of the solutions, the existence of periodic solutions and
the stability of equilibria
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