  Nonlinear reaction-diffusion systems admit a wide variety of spatiotemporal
patterns or structures. In this lecture, we point out that there is certain
advantage in studying discrete arrays, namely cellular neural/nonlinear
networks (CNNs), over continuous systems. Then, to illustrate these ideas, the
dynamics of diffusively coupled one and two dimensional cellular nonlinear
networks (CNNs), involving Murali-Lakshmanan-Chua circuit as the basic element,
is considered. Propagation failure in the case of uniform diffusion and
propagation blocking in the case of defects are pointed out. The mechanism
behind these phenomena in terms of loss of stability is explained. Various
spatiotemporal patterns arising from diffusion driven instability such as
hexagons, rhombous and rolls are considered when external forces are absent.
Existence of penta-hepta defects and removal of them due to external forcing is
discussed. The transition from hexagonal to roll structure and breathing
oscillations in the presence of external forcing is also demonstrated. Further
spatiotemporal chaos, synchronization and size instability in the coupled
chaotic systems are elucidated.
