Kuramoto-Sakaguchi Oscillators

Phase synchronization with coupling lag β — chimera states emerge

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The Kuramoto-Sakaguchi model: dθᵢ/dt = ωᵢ + (K/N)Σⱼ sin(θⱼ−θᵢ−β). The phase lag β breaks the sin symmetry, enabling traveling waves and chimera states where synchronized and desynchronized populations coexist. The order parameter r = |N⁻¹Σ e^{iθⱼ}| measures global coherence: r=1 (locked), r≈0 (incoherent).