Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators

TL;DR

This paper demonstrates that large systems of globally coupled oscillators exhibit low-dimensional dynamics in the infinite size limit, providing explicit equations and solutions for their macroscopic behavior, including extensions and time delays.

Contribution

The authors derive explicit finite-dimensional nonlinear equations describing the macroscopic dynamics of large coupled oscillator systems, including exact solutions for specific cases.

Findings

Low dimensional behavior observed in large oscillator systems

Explicit equations derived for macroscopic evolution

Exact solutions obtained for the Kuramoto model with Lorentzian distribution

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.