Impact of global interaction and symmetry on pattern selection and bifurcation

Abstract

Global interaction refers to a nonlocal mode of information exchange (coupling) between the local states on a surface. Global interaction may produce a very rich class of spatiotemporal patterns. A system has an inversion symmetry if both φ(x,y,λ) and φ(−x,−y,−λ) are solutions. Here x and y are the two dynamic variables of the system and λ is a global control variable. The presence of inversion symmetry sharpens the distinction among the various motions and leads to bifurcation scenarios which have not been found in its absence. A heteroclinic connection between two inversion symmetric saddle foci leads to many shifts between back-and-forth and unidirectional pulse branches of solutions. The scenario by which the periodic orbits gain and lose stability via period-increasing or saddle-node bifurcations is similar to one predicted by Glendining for a system described by three ordinary differential equations having inversion symmetry. The dynamic features are robust and rather insensitive to the functional form of the kinetic expression.