Chaos Synchronization — Master-Slave Lorenz Systems

Pecora-Carroll 1990: two chaotic systems with identical parameters synchronize via coupling — ‖x₁−x₂‖→0

Lorenz Parameters

σ (Prandtl): 10.0 ρ (Rayleigh): 28.0 β: 2.67

Coupling

Coupling K: 2.0 Initial separation d₀: 5.0

‖e‖ = —

State: —

Pecora-Carroll 1990: Slave drives on x-coordinate. Conditional Lyapunov exponents of slave subsystem must all be negative for synchronization. Lorenz x-coupling: λ₂+λ₃<0 always → sync guaranteed for K>0. Applications: secure chaos communications (Cuomo-Oppenheim 1993).