Phase Dynamics, Phase Resetting, Correlation Functions and Coupled Map Lattices

Some of the richest dynamical phenomena occur in both space and time. In view of the substantial developments that have taken place in the understanding of the onset of chaos in spatially homogeneous systems, which are described by systems of ordinary differential equations, there is some hope that parallel developments may take place in the study of spatially distributed systems that are usually described by partial differential equations. It is safe to say that this goal has not yet been achieved. The problem is twofold: not only are the phenomena diverse so that a set of organizing features like the major routes to chaos in spatially homogeneous systems has yet to be discovered, but some of the most useful nonlinear dynamics techniques, like surface of section plots, fractal dimensions and Lyapunov numbers, lose some of their utility for these high-dimensional systems. While all of the above tools can and have been brought to bear on the problem of spatio-temporal structure there is no one definitive diagnostic method.

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