Pattern formation in active oscillatory media and its relation to associative memory networks

We continue to study the arrays of nonlinear coupled oscillators. The networks of associative memory based on limit-cycle oscillators connected via complex-valued Hermitian matrices were previously designed. Another class of networks consisting of locally connected nonlinear oscillators, closely related to so-called cellular neural networks, is the subject of study in the present paper. In spatially continual limit these oscillatory networks can be considered as oscillatory media governed by the system of reaction-diffusion equations. Formation of spatio-temporal dissipative structures (wave trains, standing waves, targets and shock structures, spiral waves, stripe patterns, cluster states) in various nonlinear active media is widely used for modelling of complicated nonlinear phenomena in physics, chemistry, biology, neurophysiology. Here the results of analytical study of 1D oscillatory media corresponding to closed and unclosed chains of limit-cycle oscillators are presented. Conditions of existence of some spatio-temporal regimes inherent to nonlinear active media (diffusion instability caused by coupling, formation of wave trains and standing waves) have been clarified.