Spatial and temporal structure in systems of coupled nonlinear oscillators

The spatial and temporal bifurcations of a ring of coupled, discrete-time, nonlinear oscillators are studied. The model displays many of the phenomena observed in diffusively coupled, nonlinear, chemical oscillators which can possess complex dynamics when isolated. The low-order bifurcation diagram of the discrete-time model may be computed analytically and shows how in-phase and out-of-phase solutions arise and undergo further bifurcations to quasiperiodic or chaotic states. Spatial bifurcations (pattern formation) accompany the temporal bifurcations and the results indicate how some of these processes occur. The phase diagram possesses self-similar scaling features associated with the higher-order periodic states. The model should prove useful in identifying the analogous phenomena in physical systems.