Below Kc the population stays incoherent — no shared rhythm survives the spread. Above Kc, a synchronized cluster spontaneously forms and grows.
What do you see when you pass the threshold? Chaos or Order?
Every oscillator here is identical — same frequency, same coupling rule. Yet the population tears into a coherent band and an incoherent one, and nothing in the setup says where the tear goes.
Same units, same law — so why don't they all agree? Hit Re-run and watch the split move.
Kuramoto Equation
dθᵢ/dt = ωᵢ + (K/N) Σⱼ sin(θⱼ − θᵢ)
Each oscillatorOne rhythmic unit in the population — a single dot on the circle, running at its own preferred speed until coupling pulls it toward the rest. has a natural frequencyNatural frequency ωᵢ — how fast the oscillator prefers to run on its own, with no coupling. Drawn from a Lorentzian distribution centered at ω₀. Dot color tracks each oscillator's offset from the mean phase ψ — aligned dots share one hue. ωᵢ and a current phasePhase θᵢ — where the oscillator sits in its cycle, measured as an angle in [0, 2π). Each dot's position on the unit circle is its current phase. θᵢ. The coupling term pulls each oscillator toward the rest, weighted by how out-of-phase they are.
Mean-field rewrite
r · eiψ = (1/N) Σⱼ eiθⱼ
dθᵢ/dt = ωᵢ + K · r · sin(ψ − θᵢ)
The N² pairwise sum collapses to a single average. Compute the order parameterOrder parameter r — the magnitude of the mean phase vector. r ≈ 0 when phases scatter (rainbow on the circle); r ≈ 1 when they lock (one hue dominates). The bar in the bottom-left tracks r live; ψ is the locked cluster's mean phase. r and mean angle ψ once per step, reuse for all N oscillators — O(N) instead of O(N²).
Live state
r = 0.000
ψ = 0.00 rad
K = 1.00
Kc = 2.00
For a Lorentzian with half-width γ, the critical couplingCritical coupling K_c — the threshold above which synchronization spontaneously emerges. Derived from K_c = 2/(π · g(ω₀)) where g is the frequency density at the center. For a Lorentzian, g(ω₀) = 1/(πγ), giving K_c = 2γ exactly. is Kc = 2γ. Drag K above Kc to trigger the transition.
Phase transition
Below Kc, phases scatter uniformly and r fluctuates near 1/√N (finite-size noise). Above Kc, a cluster forms and revolves at the mean locked frequency; r rises toward 1. This is a phase transitionPhase transition — a sharp change in collective behavior at a critical parameter. Below K_c: disorder. Above K_c: order. Same mathematical structure as a ferromagnet becoming magnetic. driven by mean-fieldMean-field — each oscillator interacts with the collective average rather than with each individual neighbor. Exact for all-to-all coupling in the N → ∞ limit. coupling. No oscillator decides to synchronize — it emerges from the local rule.
Chimera Equation
dθᵢ/dt = ω − (1/N) Σⱼ [1 + A·cos(xᵢ − xⱼ)] · sin(θᵢ − θⱼ + α)
Every oscillator shares one natural frequencyIdentical ω — in chimera mode there is no frequency spread. All N oscillators want to run at exactly the same speed. Any difference in their behaviour is created by the dynamics, not baked into the population. ω. Coupling is non-localNon-local coupling — strength depends on how far apart two oscillators sit on the ring, set by the kernel 1 + A·cos(xᵢ − xⱼ). Neighbours pull harder than distant units. This is the structural ingredient chimeras need; all-to-all (mean-field) coupling cannot produce them in phase-only models like this one.: each unit sits at a ring position xᵢ = 2πi/N and feels its neighbours more strongly than distant units.
Kernel & phase lag
G(x) = 1 + A·cos(x)
α = π/2 − β
A sets the coupling range — how sharply near beats far. β is the phase lagPhase lag β — a small offset inside the sine, α = π/2 − β. Chimeras live in a narrow wedge of (A, β). Push β up and the incoherent region heals into full synchrony; pull it toward zero and the locked band erodes. The default sits inside the chimera band.. The chimera regime is a narrow band — the defaults sit inside it; nudge β to find its edges.
Live state
r = 0.000
A = 0.950
β = 0.10
Global r sits at an intermediate plateau — neither 0 nor 1. Part of the population locks while the rest stays incoherent, so the population-wide order parameter lands in between and stays there. That steady in-between value is itself the chimera signature.
The re-run fingerprint
Identical units, identical rule — yet the population splits into a coherent band and an incoherentSymmetry breaking — nothing in the setup distinguishes one oscillator from another, so nothing predicts which ones end up incoherent. The split is chosen by the initial conditions alone. This is what makes a chimera a chimera and not just partial synchronization. band. Nothing in the parameters says where. Hit Re-run and the boundary in the spatial strip lands somewhere new every time — that relocation is the proof the split is born from the dynamics, not the recipe (like Kuramoto & Battogtokh's original experiment, the sim seeds a coherent patch — chimeras coexist with full sync and must be steered into; what's not chosen is that the split holds).
About Yoshiki Kuramoto →