Fractal evolution of normalized feedback systems on a lattice

Abstract

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 [G. Grössing and A. Zeilinger, Complex Syst. 2 (1988) 197, 611; Physica D 31 (1988) 70] 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 system variables, the dynamical properties can be characterized by a “fractal evolution dimension” (as opposed to a “fractal dimension”).