Coupled Map Lattices

Coupled map lattices (CMLs) are spatially extended dynamical systems consisting of a lattice of discrete-time maps, each evolving according to a local rule, with coupling to neighboring sites. Introduced by Kaneko in the 1980s, CMLs occupy a middle ground between cellular automata (discrete state space, discrete time) and partial differential equations (continuous state space, continuous time). They are the minimal model for studying how local chaos and spatial coupling interact to produce collective behavior — pattern formation, synchronization, turbulence, and phase transitions.

A CML is defined by two equations: a local map (typically chaotic, such as the logistic map) and a coupling term (typically diffusive, averaging each site with its neighbors). Despite the simplicity of these components, CMLs exhibit remarkably complex behavior. Weak coupling preserves local chaos; strong coupling produces spatial coherence. At intermediate coupling, the system enters a regime of spatiotemporal