Fractal evolution in deterministic and random models

One-dimensional coupled map lattices or quantum cellular automata with any additionally implemented temporal feedback operations (involving some memory of the system's states) and a normalization procedure after each time step exhibit a universal dynamic property called fractal evolution [Fussy & Grossing, 1994]. It is characterized by a power-law behavior of a system's order parameter with regard to a resolution-like parameter which controls the deviation from an undisturbed (i.e., feedback-less) system's evolution and provides a dynamically invariant measure for the emerging spatiotemporal patterns. By comparison with another, simpler model without memory, where the patterns are generated randomly, the underlying principles of fractal evolution are studied. It is shown that our system evolving entirely deterministically, exhibits properties occurring usually only in random models, where the global measures, up to a certain degree, are calculable. Other properties like the fractal evolution exponent remain in general computationally irreducible due to the self-referential feedback dynamics. A specific model with an approximate estimation of the fractal evolution exponent is discussed. The stability of fractal evolution with respect to the dependence of pattern formation on the systems variables is also analyzed.