Easily calculable measure for the complexity of spatiotemporal patterns

We demonstrate by means of several examples that an easily calculable measure of algorithmic complexity c which has been introduced by Lempel and Ziv [IEEE Trans. Inf. Theory IT-22, 25 (1976)] is extremely useful for characterizing spatiotemporal patterns in high-dimensionality nonlinear systems. It is shown that, for time series, c can be a finer measure for order than the Liapunov exponent. We find that, for simple cellular automata, pattern formation can be clearly separated from a mere reduction of the source entropy and different types of automata can be distinguished. For a chain of coupled logistic maps, c signals pattern formation which cannot be seen in the spatial correlation function alone.