Formation and propagation of localized states in extended systems

We study a one-species-diffusion equation supplemented by a nonlinear saturation term. If this term includes temporal delay, a rich spatio-temporal behavior already at the critical point where the steady homogeneous solution gets unstable is expected. Normal form transformation into a complex Ginzburg-Landau equation shows that the condition for phase turbulence is always fulfilled. As a secondary instability of these chaotic states well ordered spiral waves and targets are found numerically. Here the system shows features well-known from excitable media. Finally we propose the construction of logical circuits by means of a network built up by one-dimensional excitable media. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.