Experiment C · Edge-of-chaos diagnostic · Lyapunov sweep

12 coupled phases.
Find the boundary where it thinks fastest.

Two trajectories start ε apart in phase space. The larger the growth in their separation per unit time, the larger the maximum Lyapunov exponent. Where λ crosses zero is where chaos becomes order — the most information-rich regime in either direction.
EXPLAIN LIKE I'M 10
· everyday-language one-paragraph version
Twelve metronomes again — but this time the question isn't 'will they sync', it's 'how chaotic are they'. Turn the coupling knob low and the chain runs wild. Turn it high and it falls into boring perfect sync. Right between those two is the interesting spot — that's where the system is 'thinking' fastest, holding the most information per second. The page finds that spot for you and shows the curve.
WHAT YOU ARE LOOKING AT
A chain of N coupled phase oscillators. Each has its own natural frequency. The coupling strength K controls how much they pull on each other. At low K they run incoherent — chaotic. At high K they synchronise — ordered. The page sweeps K and tracks two diagnostic numbers: the Lyapunov exponent λ (sensitivity to a tiny perturbation) and the order parameter r (how aligned the oscillators are).
WHY THIS MATTERS FOR QUANTUM
The 'edge of chaos' — the regime where λ is near zero, r fluctuates, and the system is neither ordered nor random — is precisely where physical systems are best at information processing. Quantum scrambling, OTOC growth, eigenstate-thermalisation onset all happen at this boundary. Hardware operated at edge of chaos has the largest information capacity for a given energy budget. The simulator finds where that K-window sits for the current chain.
WHAT TO WATCH
Drive the K slider from 0.1 to 4.0. λ starts positive (chaotic), drops through zero (edge), goes negative (ordered). Read the regime label in the headline. Slide the spread slider to test how disorder in the natural frequencies moves the edge. The K_c boundary is sensitive to disorder; the edge is where the geometry's own resonance lives.
K · coupling
1.00
spread Δω=1.00
Lyapunov λ
-0.031
contractive
Order parameter r
0.598
fluctuating
Regime
edge
maximum information processing
Order parameter r(t) · current K
copper = sync (r→1) · steel = mid · null = chaotic
Lyapunov λ(K) · 24-point sweep
Press Run sweep →
The physics, plainly

N phase oscillators coupled all-to-all via the Kuramoto interaction dθ_i/dt = ω_i + (K/N) · Σ sin(θ_j − θ_i). Two-trajectory FTLE. The Lorentzian-distributed natural frequencies (spread Δω) define an analytic critical coupling K_c = 2 · Δω / π. Above K_c the global attractor is synchronised; below it the chain is chaotic. The page measures both regimes empirically and identifies the edge.

Quantum systems poised at edge-of-chaos exhibit maximum out-of-time-ordered correlator (OTOC) growth, fastest scrambling, and the largest Lyapunov-like growth rate of information. Operating quantum hardware in this regime maximises information capacity per unit power — a different design principle than cold-and-isolated.

At the current K = 1.00, the chain is edge. Predicted K_c for this Δω: 0.64.