Spatiotemporal Chaos — Coupled Logistic Maps

A lattice of logistic maps coupled to their neighbors generates rich spatiotemporal dynamics: from synchrony to defect turbulence and coherent structures. The space-time diagram reveals the coexistence of laminar regions and chaotic bursts characteristic of spatiotemporal intermittency.

Logistic r: 3.8 Coupling ε: 0.3 Lattice size L: 80 Color scheme:

Time step: 0

Max Lyapunov: —

Mean x: —

Variance: —

xᵢ(t+1) = (1-ε)·f(xᵢ) + ε/2·[f(xᵢ₋₁)+f(xᵢ₊₁)]

f(x) = r·x·(1−x)

ε=0: independent chaos
ε=1: fully coupled

Regimes: r=3.8, ε=0.3: spatiotemporal chaos; ε→0: independent maps; ε=0.5, r=3.5: traveling waves. Space-time diagram: x-axis=site, y-axis=time (down), color=x value. Kaneko (1989) first systematically studied CML dynamics.