Chaos Synchronization — Pecora-Carroll Coupling

Conditional Lyapunov exponents λ₂,λ₃ < 0 → synchronization; coupling strength ε controls onset

System type

Coupling ε 0.30

Coupling type

σ (Lorenz) 10.0

ρ (Lorenz) 28.0

β (Lorenz) 2.67

Pecora-Carroll (1990): Drive-response (master-slave) synchronization. The master system drives a sub-system of the slave. If the conditional Lyapunov exponents of the response subsystem are all negative, synchronization occurs: |x_master − x_slave| → 0. Coupling ε must exceed critical threshold ε_c. Lorenz system: ẋ=σ(y−x), ẏ=x(ρ−z)−y, ż=xy−βz. Chaotic for σ=10,ρ=28,β=8/3. Diffusive coupling: ẋ_s = f(x_s) + ε(x_m − x_s). Colors: Lorenz butterfly wings encoded as hue.