Frustrated Kuramoto Oscillators

N coupled phase oscillators with random natural frequencies and frustration phase α. Partial synchronization, chimera-like coexistence, and rich order-parameter dynamics emerge from competition between coupling and disorder.

Phase Circle (oscillator positions)

Order Parameter r(t)

Oscillators N 30

Coupling K 1.5

Freq spread σ 1.0

Frustration α 0.00

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Order r

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Mean phase ψ

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Kc (theory)

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Regime

Frustrated Kuramoto Model

dθᵢ/dt = ωᵢ + (K/N) Σⱼ sin(θⱼ − θᵢ − α)

Order parameter: r·e^(iψ) = (1/N) Σ e^(iθⱼ)

Critical coupling: Kc = 2σ√(2/π) (Lorentzian ω dist.)

α≠0 → frustration: synchrony coexists with drift

α=π/2 → conserved quantity, Hamiltonian limit