Kuramoto Chimera State

Nonlocal coupling on a ring — spontaneous coexistence of synchrony and incoherence

OSCILLATOR RING (phase as color, 0=red, π=cyan)

LOCAL ORDER PARAMETER r(i)

PHASE DISTRIBUTION θ(i)

GLOBAL ORDER R(t)

Coupling K 2.0

Coupling range σ 0.35

α (phase lag) 1.46

Global R: –

Sync fraction: –

Time: 0

Chimera states (Kuramoto & Battogtokh 2002, named by Abrams & Strogatz 2004): in a ring of identical Kuramoto oscillators with nonlocal coupling G(x) = exp(−σ|x|)/(2σ), both synchronized and incoherent oscillator populations can coexist indefinitely — despite identical natural frequencies and symmetric initial conditions. The coupling kernel is cos(θ_j − θ_i − α) with phase lag α ≈ π/2 − 0.1. Key parameters: K (coupling strength), σ (coupling range, smaller = more nonlocal). The phenomenon requires a careful initial condition — try the chimera IC button for a half-synchronized start.