Order, chaos and complexity in discrete dynamical systems

A method for analyzing discrete dynamical systems is presented that provides a unified quantitative description of order, chaos and complexity in terms of information flow across system boundaries. Complexity is identified with variability in the relative dominance of order and chaos as systems evolve in time; therefore, purely ordered or purely chaotic behavior is considered simple. This notion of complexity is quantitatively expressed as fluctuation in net information gain. The method is applied to one-dimensional cellular automata, which are spatially and temporally discrete systems. Evidence is presented for a correlation between information fluctuation and the existence of internally complex propagating structures known as gliders. Gliders have been used in the construction of computing machines within cellular automata. This indicates that information variables may provide a connection between dynamical and computational notions of complexity. The method is also applied to one-dimensional maps, which are temporally discrete but spatially continuous, by partitioning the spatial dimension. For the logistic map, information fluctuation is maximum at the threshold between ordered and chaotic behavior, in agreement with the results of other researchers