2,390 research outputs found
Dynamics of the Chain of Oscillators with Long-Range Interaction: From Synchronization to Chaos
We consider a chain of nonlinear oscillators with long-range interaction of
the type 1/l^{1+alpha}, where l is a distance between oscillators and 0< alpha
<2. In the continues limit the system's dynamics is described by the
Ginzburg-Landau equation with complex coefficients. Such a system has a new
parameter alpha that is responsible for the complexity of the medium and that
strongly influences possible regimes of the dynamics. We study different
spatial-temporal patterns of the dynamics depending on alpha and show
transitions from synchronization of the motion to broad-spectrum oscillations
and to chaos.Comment: 22 pages, 10 figure
Fractional dynamics of coupled oscillators with long-range interaction
We consider one-dimensional chain of coupled linear and nonlinear oscillators
with long-range power-wise interaction. The corresponding term in dynamical
equations is proportional to . It is shown that the
equation of motion in the infrared limit can be transformed into the medium
equation with the Riesz fractional derivative of order , when
. We consider few models of coupled oscillators and show how their
synchronization can appear as a result of bifurcation, and how the
corresponding solutions depend on . The presence of fractional
derivative leads also to the occurrence of localized structures. Particular
solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear
Schrodinger) equation are derived. These solutions are interpreted as
synchronized states and localized structures of the oscillatory medium.Comment: 34 pages, 18 figure
Fractional dynamics of systems with long-range interaction
We consider one-dimensional chain of coupled linear and nonlinear oscillators
with long-range power wise interaction defined by a term proportional to
1/|n-m|^{\alpha+1}. Continuous medium equation for this system can be obtained
in the so-called infrared limit when the wave number tends to zero. We
construct a transform operator that maps the system of large number of ordinary
differential equations of motion of the particles into a partial differential
equation with the Riesz fractional derivative of order \alpha, when 0<\alpha<2.
Few models of coupled oscillators are considered and their synchronized states
and localized structures are discussed in details. Particularly, we discuss
some solutions of time-dependent fractional Ginzburg-Landau (or nonlinear
Schrodinger) equation.Comment: arXiv admin note: substantial overlap with arXiv:nlin/051201
- …