Logistic maps coupled diffusively — kymograph reveals turbulent, laminar, and frozen phases
x_i(t+1) = (1−ε)·f(x_i(t)) + (ε/2)·[f(x_{i-1}) + f(x_{i+1})] | f(x) = r·x·(1−x)
Coupled Map Lattice (Kaneko 1984): A spatially extended dynamical system where discrete-time chaotic maps are coupled to neighbors. Simple to simulate, yet produces rich spatiotemporal behavior: fully turbulent chaos, frozen random patterns, pattern selection, and defect turbulence.
Phase diagram: For logistic CML — at low ε, each site behaves independently (turbulent); at intermediate ε, spatial patterns form; at high ε, frozen patterns or periodic orbits emerge. The kymograph (space-time plot) shows characteristic diagonal streaks during chaos.
Lyapunov spectrum: The maximum Lyapunov exponent λ_max > 0 indicates spatiotemporal chaos. λ = ln(r|1−2x̄|)·(1−ε) for uniform state. Coupled systems develop a Lyapunov spectrum with L positive exponents — extensive chaos with dimension proportional to system size.