Phenomenology of Spatial-Temporal Chaos

ABSTRACT: The diversity of pattern generation in spatially-extended systems is investigated with lattice dynamical systems. These consist of local discrete-time dynamical systems coupled in a spatial lattice. Examples with one spatial dimension are discussed in detail, including period-doubling and phase lattices with bidirectional and unidirectional coupling. These lattices exhibit a wide range of spatio-temporal behavior including: intermittency in space and time; domains and walls; spatial period-doubling; pattern competition; spatial quasi-periodicity; soliton propogation, interaction with phase shift, annihilation, and turbulence; and exponentially long-lived transient spatial chaos. These examples support the contention that complexity in spatially-extended dynamical systems is governed by two deterministic mechanisms: the temporal amplification of information and its spatial transmission.

The rich phenomenology is illustrated with temporal, spatial, and space-time return maps, space-time diagrams, and site and pattern bifurcation diagrams. Special topics include the spatial self-similarity of patterns and finite-size scaling of periodic behavior and transients. As appropriate, analytical properties are presented.