ABSTRACT
Many systems of interest in physics, chemistry and the biological sciences exhibit spontaneous symmetry-breaking instabilities that form structures that we may call patterns [14, 54]. The appearance of convection cells in a fluid layer heated from below [81] or of vortex structures in the flow between two independently rotating cylinders [22, 81] are familiar examples from fluid mechanics. But there are many other systems that form patterns. Recently studied examples of spatially periodic structures include the Turing instability [110] and the Faraday system [83, 101]. Vertically vibrated granular media exhibit similar patternforming instabilities [99, 124]. Related instabilities have been identified in the primary visual cortex and may be responsible for hallucinations (see chapter 11 by Bressloff and Cowan). Other systems form spiral waves or target patterns, emanating from apparently random locations, perhaps triggered by impurities or specks of dust [61]. A remarkable recent discovery is that of oscillons, or localized oscillations, in vertically vibrated granular media [156]. These oscillons may form ‘bound states’ that have been called dimers. Certain chemical systems break up into spots which grow, and fission into new spots or disappear depending on the density of spots around them [88,112]. Such pattern-forming systems thus behave almost like a colony of living organisms.
