The Kuramoto Model - Introduction to Synchronization

Synchronization of Globally Coupled Nonlinear Oscillators
The Rich Behavior of the Kuramoto Model

Bryan Daniels
Spring 2005

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Introduction to Synchronization

An example of spontaneous synchronization:

To demonstrate the phenomenon of spontaneous synchronization, I set up five metronomes on a strip of balsa wood, and set them on top of two aluminum cans on their side. This setup allows to metronomes to "feel" each other through motions of the board. Click the link to see a video, and watch what happens!

Synchronization of Metronomes Video (.avi, 3.7 mb)

The phenomenon of synchronization is all around us. It is found in:

Circadian rhythms

Many living organisms synchronize to the day-night cycle.

Electrical generators

All of the generators producing power on a power grid must be synchronized to one another.

Josephson junction arrays

Josephson junctions are fabricated, microscopic devices that behave as compact electronic oscillators, and different geometric arrangements of Josephson junctions form systems of oscillators that have been found theoretically to exhibit synchronization.

Heart, intestinal muscles

The muscles in your heart, for example, must all be synchronized to create a coherent beat.

Menstrual cycles

It has been found that the menstrual cycles of women who live together often synchronize.

Fireflies

Certain species of fireflies have been found to synchronize, creating swaths of light turning on and off at the same time.

Applause

In Eastern Europe, it is customary to clap in synchrony after a good performance.

In my research, I looked at self-sustained oscillators. A self-sustained oscillator:

Oscillates in a stable pattern when left to itself;
Has certain intrinsic natural frequency;
Has a phase that tells us where it is in its cycle.

A self-sustained oscillator at three levels of abstraction. First, the real physical oscillator, in this case a metronome. Second, the limit cycle in phase space. Third, we have abstracted away everything about the oscillator except its natural frequency, its phase, and the time derivative of its phase.

With a large group of oscillators, spontaneous synchronization can happen with:

Small frequency detuning

For synchronization, a smaller range of natural frequencies is better.

Large coupling

The interactions between the oscillators must be strong enough to pull them into synchrony.

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