James P. Crutchfield
Physics Department
University of California
Berkeley, CA 94720, USA
Kunihiko Kaneko
Institute of Physics
Caollege of Arts and Sciences
University of Tokyo
Tokyo 153, JAPAN
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.
B. A. Huberman and J. P. Crutchfield. "Chaotic States of Anharmonic Systems in Periodic Fields." Phys. Rev. Let. 43 1979: 1743-1747.