Synchronization of Globally Coupled Nonlinear Oscillators
The Rich Behavior of the Kuramoto Model
An example of spontaneous synchronization:
- To demonstrate the phenomenon of spontaneous synchronization, I
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
each other through motions of the board. Click the link to see a
video, and watch what happens!
The phenomenon of synchronization is all around us. It is
- Circadian rhythms
- Many living organisms synchronize to the day-night
- 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
- 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.
- Certain species of fireflies have been found to
synchronize, creating swaths of light turning on and off at the same
- 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
- 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.
Next Section: The Kuramoto Model