Chooy. Solrrons & Fracralr Vol. 4. No. 7, pp 1193-1209. 1994 Coovrieht C? 1994 Elrewer Science Ltd Pergamon Printed m Great Britain. All rights rcscrved 0960-0779/94%7.00 + .OO Spatio-Temporal Chaos in Closed and Open Systems J. BRINDLEY Department of Applied Mathematical Studies and Centre for Nonlinear Studies, University of Leeds, Leeds LS2 9JT, UK K. KANEKO Department of Pure and Applied Sciences, College of Arts and Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo, 153 Japan T. KAPITANIAK Division of Control and Dynamics, Technical University of Lodz, Stefanowskiego l/15, 90-924 Lodz. Poland zyxwvutsrqpon (Received 28 July 1993) Abstract--patio-temporal chaos (temporal chaos coupled with spatial variability) widely occurs in turbulent phenomena and is associated with spatial pattern formation. In this review we address classical examples of spatio-temporal complexity, develop ideas in the context of coupled map lattices and speculate on possible quantifiers for spatio-temporal chaos. 1. INTRODUCTION: WHAT IS SPATIO-TEMPORAL CHAOS? By spatio-temporal chaos (STC) we mean temporal high-dimensional chaos associated with spatial pattern dynamics. It occurs widely in turbulent phenomena, including Rayleigh- Bernard convection, electrically driven convection in liquid crystals, boiling, combustion, MHD turbulence in plasma, solid-state physics (Josephson junctions, spin wave turbu- lence), optics, chemical reactions with spatial structure, and so on. It is also important in biological information processing involving nonlinear characteristics, for example, neural dynamics. Although there is no clear definition for STC, we assume that the number of degrees-of-freedom is large and that the dimension (in phase-space) increases with the system’s size. In the purely temporal case, studies on low-dimensional chaos have expanded rapidly, and some understanding has developed. Dynamics of ordered spatial structure has been studied in pattern formation. However, though phenomena complex in both space and time are common in nature, little basic understanding has yet been developed. We are at an exploratory stage, seeking new phenomenology in a jungle of spatio-temporal chaos. Understanding the phenomenology may still require much time, but we present evidence in Section 5 of a structured hierarchy of qualitative behaviour, which gives support to the idea of at least some universality classes of STC. As a preliminary, we review briefly in Section 2 some of the methods of recognizing and qualifying temporal chaos which may also be of value for STC. In Section 3 we briefly 1193