Fractional dynamics of coupled oscillators with long-range interaction.

Vasily E Tarasov; George M Zaslavsky

doi:10.1063/1.2197167

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Fractional dynamics of coupled oscillators with long-range interaction.

Vasily E Tarasov·George M Zaslavsky

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Abstract

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.

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Vasily E Tarasov, George M Zaslavsky. 2006. Fractional dynamics of coupled oscillators with long-range interaction.. https://doi.org/10.1063/1.2197167

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