Pecora-Carroll Chaos Synchronization

Two Lorenz oscillators — driver and response — synchronizing through a unidirectional signal
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Sync error |x₁−x₂| (avg)

Chaos Synchronization

Pecora and Carroll (1990) showed that chaotic systems can synchronize when one (the driver) feeds into a response system — despite sensitive dependence on initial conditions.

ẋ₁ = σ(y₁−x₁) ẏ₁ = x₁(ρ−z₁)−y₁ ż₁ = x₁y₁−βz₁
ẋ₂ = σ(y₂−x₂) − k(x₂−x₁)

The coupling term drives x₂ → x₁. Lyapunov exponents of the response system become negative — the transverse manifold is stable.

Try coupling = 0 to see divergence; increase it to watch synchronization lock in. The scatter plot below shows x₁ vs x₂ collapsing to the diagonal.