Spatiotemporal patterns in arrays of coupled nonlinear oscillators

Abstract

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.