Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators

Abstract

It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.

resubmission of Tex files

References in corpus (4)

• Synchronization in networks of networks: the onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators
• The Spectrum of the Partially Locked State for the Kuramoto Model
• External Periodic Driving of Large Systems of Globally Coupled Phase Oscillators
• Time delay in the Kuramoto model with bimodal frequency distribution

Cited by in corpus (8)

• Exact Results for the Kuramoto Model with a Bimodal Frequency Distribution
• Partially integrable dynamics of hierarchical populations of coupled oscillators
• Chimera states in heterogeneous networks
• Stability diagram for the forced Kuramoto model
• External Periodic Driving of Large Systems of Globally Coupled Phase Oscillators
• Invariant submanifold for series arrays of Josephson junctions
• Low Dimensional Description of Pedestrian-Induced Oscillation of the Millennium Bridge
• Echo Phenomena in Large Systems of Coupled Oscillators