Kuramoto Model
The Kuramoto model is a mathematical model that describes how a group of coupled oscillators can synchronize their phases despite having different natural frequencies. It was proposed by Japanese physicist Yoshiki Kuramoto (蔵本由紀) in the 1970s and is widely used in physics, biology, neuroscience, engineering, and sociology to study collective synchronization phenomena.
Basic mathematical definition
Suppose you have \(N\) oscillators, each with a phase \(\theta_{i}(t)\) that evolves over time. The dynamics are given by:
$$\frac{d\theta_{i}}{dt} = \omega_{i} + \frac{1}{N}\sum_{j = 1} K_{ij} \sin \left( \theta_{j} - \theta_{i} \right)$$
where:
- \(\theta_{i}\): phase of oscillator \(i\).
- \(\omega_{i}\): natural frequency of oscillator \(i\).
- \(K\): coupling constant (interaction strength between oscillators).
Model intuition
- Each oscillator tends to follow its own natural frequency \(\omega_i\).
- At the same time, each oscillator is influenced by the others (the sine term), which tends to bring their phases closer together.
- The balance between the spread of natural frequencies and the coupling strength \(K\) determines whether the system synchronizes.
Key results
- When \(K\) is small, each oscillator follows its own frequency, and the system remains unsynchronized.
- If \(K\) exceeds a certain critical threshold \(K_{c}\), the system can begin to synchronize: a group of oscillators starts to move in phase (partial synchronization).
- In the limit as \(N \to \infty\), the model can be analyzed using integral equations to obtain statistical solutions.
Simulation
In this simulation, you can visualize the behavior of the Kuramoto model with a collective signal and the synchronization parameter \(r(t)\). The simulation shows how the oscillators evolve over time, how they synchronize, and how the collective signal behaves. It represents oscillators as points on a circle. Each oscillator has a phase \((\theta)\) and a natural frequency \((\omega)\).
Instructions
Adjust the coupling constant \(K\) to observe the synchronization phenomenon. The sum of the sinusoidal components is calculated and graphed:
$$S\left( t \right) = \frac{K}{N} \sum_{j = 1}^{N} \sin \left( \theta_{j}(t) \right)$$
This sum is plotted as a signal on an oscilloscope-like graph below, \(r(t)\) is the Kuramoto order parameter that measures the degree of system synchronization:
$$r(t) = \left\vert \frac{1}{N} \sum_{j = 1}^{N}e^{i\theta_{j}(t)} \right\vert$$
Values of \(r\) near \(1\) indicate high synchronization, and values near \(0\) indicate complete phase shift.
See also
Fourier transform