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Fractional dynamics of coupled oscillators with long-range interaction
Vasily E Tarasov1, George M Zaslavsky
1Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia.
Chaos (Woodbury, N.Y.)
|July 11, 2006
Summary
This study explores coupled oscillators with long-range interactions, revealing synchronization and localized structures via fractional derivatives. The findings link Riesz fractional derivatives to phenomena in nonlinear dynamical systems.
Area of Science:
- Nonlinear Dynamics
- Fractional Calculus
- Condensed Matter Physics
Background:
- Coupled oscillators exhibit complex behaviors, including synchronization.
- Long-range interactions in physical systems can lead to unique emergent properties.
- Fractional calculus offers a powerful framework for modeling anomalous dynamics.
Purpose of the Study:
- To investigate synchronization and localized structures in a 1D chain of coupled oscillators with power-law interactions.
- To analyze the role of Riesz fractional derivatives in the infrared limit of these systems.
- To derive and interpret solutions for fractional nonlinear equations.
Main Methods:
- Modeling a 1D chain of coupled linear and nonlinear oscillators with power-law interactions.
- Transforming the equation of motion in the infrared limit to incorporate Riesz fractional derivatives (0 < alpha < 2).
- Analyzing synchronization through bifurcation theory and deriving particular solutions for fractional complex Ginzburg-Landau/nonlinear Schrödinger equations.
Main Results:
- Synchronization in coupled oscillators emerges as a result of bifurcation, dependent on the fractional order alpha.
- The presence of the Riesz fractional derivative leads to the formation of localized structures.
- Specific solutions for fractional time-dependent complex Ginzburg-Landau/nonlinear Schrödinger equations were derived.
Conclusions:
- Fractional derivatives are crucial for understanding synchronization and localized structures in these oscillatory media.
- The parameter alpha significantly influences the dynamics and emergent behaviors of the coupled oscillator system.
- The derived solutions provide insights into synchronized states and localized phenomena in fractional dynamical systems.
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