🔗 Coupled Oscillators – Emergent Synchronisation

This simulation implements the Kuramoto model: a classic model of how synchronisation emerges from the interaction of many oscillators with different natural frequencies.

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🧠 Idea

• N oscillators each have their own natural frequency ω_i.
• They are coupled through pairwise sine interactions:

dθ_i/dt = ω_i + (K/N) Σ_j sin(θ_j − θ_i)

• Below a critical coupling K_c: oscillators remain incoherent (each runs at its own speed).
• Above K_c: oscillators spontaneously synchronise – a phase transition to collective order.

This is the same mechanism behind synchronising fireflies, pacemaker cells, and coupled neuronal oscillators.

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📊 Order Parameter

The simulation tracks r(t) ∈ [0, 1]:

r	Meaning
≈ 0	Fully incoherent, random phases
≈ 1	Perfectly synchronised, all phases aligned

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🖼 Visualisation

• Left panel – unit circle with oscillator phases as dots; red arrow = mean field direction and magnitude r
• Right panel – order parameter r(t) over time

Press ESC to exit.

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▶ Run

cd simulation-models/coupled-oscillators
python3 coupled_oscillators.py

Experiment ideas

• Set K_COUPLING = 0.5 → no synchronisation
• Set K_COUPLING = 4.0 → fast, strong synchronisation
• Increase N_OSCILLATORS for a cleaner phase transition

📚 References

• Kuramoto, Y. (1975). Self-entrainment of a population of coupled non-linear oscillators. International Symposium on Mathematical Problems in Theoretical Physics.