- Published 2006 in Chaos

We consider a one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction. The corresponding term in dynamical equations is proportional to 1//n-m/alpha+1. 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 alpha, when 0<alpha<2. We consider a few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on alpha. The presence of a fractional derivative also leads 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.

@article{Tarasov2006FractionalDO,
title={Fractional dynamics of coupled oscillators with long-range interaction.},
author={Vasily E. Tarasov and George Zaslavsky},
journal={Chaos},
year={2006},
volume={16 2},
pages={023110}
}