Pattern Formation in Spatially Discrete Systems

Mendez et at., Reaction-Transport Systems, Springer Series in Synergetics, DOI 10.1007/978-3-642-11443-4 13, Springer-Verlag Berlin Heidelbeig 2010 

The earliest studies of coupled chemical cells were theoretical investigations and focused on the diffusion-induced instability to nonuniform steady states [354, 443] in two coupled reactors. More recently, Epstein and Golubitsky investigated the Turing instability in linear and circular arrays of Brusselators [109]. Their approach makes use of underlying symmetry groups. 

This chapter focuses on the stability properties of networks or arrays of coupled monostable units or cells. We consider two types of coupling, namely diffusive coupling and photochemical coupling. The two main concerns are how the topology of the network connectivity and how spatial inhomogeneities in the array affect instabilities. Spatially discrete systems or networks of coupled cells are described by sets of ordinary differential equations. Methods to determine the stability of stationary states of ODEs are well developed. 

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