Fireflies in Sync: Kuramoto Coupled Oscillators |…

What Are Coupled Oscillators?

Coupled oscillators are systems where individual oscillators interact with each other. This interaction can cause them to synchronize their behavior over time.

A classic example is fireflies, which flash in unison despite being independent individuals.

The Kuramoto Model

The Kuramoto model describes how a large population of coupled oscillators with natural frequencies interact and can lead to synchronization.

It uses a phase variable for each oscillator and a coupling strength parameter to describe the interaction between them.

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Phase Circle Representation

In the context of Kuramoto oscillators, the phase circle is a useful visualization tool where each point on the circle represents an oscillator’s phase.

The position of points on this circle changes over time as the oscillators adjust their phases to match those around them.

Order Parameter and Synchronization

The order parameter, r(t), quantifies the degree of synchronization in a population of oscillators.

When all oscillators are perfectly synchronized, r approaches 1; when they are completely out of sync, r approaches 0.

Frequently asked questions

How does the natural-frequency histogram relate to the Kuramoto model?

The natural-frequency histogram shows the distribution of intrinsic frequencies among oscillators. This distribution is crucial as it determines how easily synchronization can occur.

What does the locking window in the phase circle represent?

The locking window indicates the range of phases where oscillators are most likely to synchronize, reflecting the influence of coupling strength and natural frequencies.

Why is the critical coupling K_c = 2/(π g(0)) significant?

This value marks the threshold at which a population of oscillators transitions from an unsynchronized state to a synchronized one, highlighting the importance of coupling strength in achieving synchronization.

How can we apply this model beyond fireflies?

The Kuramoto model is applicable in various fields such as neuroscience (neuron firing), power grids (synchronization of electrical signals), and even social dynamics (opinion formation).

Try it live

Everything above runs in your browser — open Fireflies in Sync: Kuramoto Coupled Oscillators — 2D Phase Circle & Order Parameter and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Fireflies in Sync: Kuramoto Coupled Oscillators — 2D Phase Circle & Order Parameter simulation