Pecora-Carroll Chaos Synchronization

Coupled Lorenz Systems & Master Stability Function
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Pecora-Carroll Theorem (1990)

Drive: ẋ₁=σ(y₁-x₁), ẏ₁=ρx₁-y₁-x₁z₁, ż₁=-βz₁+x₁y₁

Response: ẋ₂=σ(y₂-x₂), ẏ₂=ρx₁-y₂-x₁z₂, ż₂=-βz₂+x₁y₂

The response system is a copy of the drive's y-z subsystem, driven by x₁. Pecora & Carroll (1990) showed that if the conditional Lyapunov exponents of the response subsystem are all negative, complete synchronization occurs: (y₂,z₂)→(y₁,z₁). This was the first controlled chaos synchronization, enabling chaos-based communications (Cuomo & Oppenheim 1993).