Fractal Evolution of Normalized Feedback Systems on a Lattice

Highly nonlinear behaviour of a system of discrete sites on a lattice is observed when a specific feedback loop is introduced into models employing quantum cellular automata [1] or their real-valued analogues. It is shown that the combination of two operations, i.e. i) enhancement of a site’s value when fulfilling a feedback condition and ii) normalization of the system after each time step, produces relatively short-lived spatiotemporal patterns whose mean lifetime can be considered as emergent order parameter of the system. This mean lifetime obeys a scaling law involving a control parameter which tunes the“fault tolerance”of the feedback condition. Thus, within appropriate ranges of the systems variables, the dynamical properties can be characterized by a “fractal evolution dimension” (as opposed to a “fractal dimension”).