Chaos Synchronization

Two Lorenz systems with coupling κ. Below the threshold, they diverge exponentially. Above it, complete synchronization emerges — identical trajectories despite chaos.

Coupling κ 5.0 σ (Prandtl) 10.0 ρ (Rayleigh) 28.0 β 2.67

Checking...

Error |x₁−x₂|: –
Lyapunov λ₁: ≈ 0.906
Sync threshold: κ ≈ 9–10

Pecora-Carroll (1990): drive-response synchronization. Conditional Lyapunov exponents all negative → synchronization. Coupling: ẋ₁ += κ(x₂−x₁). Complete sync when κ exceeds the largest conditional Lyapunov exponent.