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Related Experiment Video

Updated: Jun 26, 2026

Magnetically Induced Rotating Rayleigh-Taylor Instability

Magnetically Induced Rotating Rayleigh-Taylor Instability

Published on: March 3, 2017

Stability diagram for the forced Kuramoto model.

Lauren M Childs¹, Steven H Strogatz

¹Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA. lmchilds@cam.cornell.edu

Chaos (Woodbury, N.Y.)

|January 7, 2009

Summary

This study analyzes the periodically forced Kuramoto model, revealing exact results for bifurcations between synchronization states. The findings clarify complex dynamics in coupled oscillator systems, with applications in physics, chemistry, and biology.

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Area of Science:

Physics
Complex Systems
Nonlinear Dynamics

Background:

The Kuramoto model describes synchronization in populations of coupled oscillators.
Mutual synchronization competes with external forcing in many natural and engineered systems.
Previous studies identified forced and mutual entrainment attractors but lacked detailed bifurcation analysis.

Purpose of the Study:

To perform a complete bifurcation analysis of the periodically forced Kuramoto model.
To clarify the transitions between different synchronization states (attractors).
To investigate a simplified two-dimensional case of the infinite-dimensional system.

Main Methods:

Mathematical analysis of the periodically forced Kuramoto model.
Reduction of infinite-dimensional dynamics to a two-dimensional system.
Exact calculation of bifurcation points (Hopf, saddle-node, Takens-Bogdanov).

Main Results:

Exact locations of Hopf, saddle-node, and Takens-Bogdanov bifurcations were determined.
A comprehensive stability diagram for the two-dimensional system was established.
The stability diagram shows strong similarity to that of the forced van der Pol oscillator.

Conclusions:

The study provides a complete bifurcation analysis for a key case of the forced Kuramoto model.
The findings elucidate the complex interplay between mutual and forced synchronization.
The results offer insights into synchronization phenomena across various scientific disciplines.