Exact solvability and quasiperiodicity of one-dimensional cellular automata

Cellular automata are a class of mathematical systems characterized by discreteness (in space, time, and state values), determinism, and local interaction. Little is known mathematically about automata with nonlinear interaction rules. This paper establishes that certain nonlinear automata on finite lattices may be mapped exactly onto a linear automaton, thus providing an 'exact solution' for the nonlinear systems, and permitting description of their fundamental dynamical features such as limit cycle period, attractor SI~UCIU~C, and transience length. These particular nonlinear automata generate multiple domains within which evolution exactly mimics that of the linear automaton, with ;he domain wall behaviour itself governed by the dynamics of the linear automaton. In particular, the position of the domain walls follows a trajectory that is determined by the linear system. and is characterized by an integer-valued 'winding number' representing the 'oe:,aviour, The :iiiii; 'oehavioiii of >he nonlinear Bil;onTa;d on finite lattices is then determined by the periodicity of the associated linear 'template' system modulated by the winding numbers of the domain walls, and hence may viewed as providing a realization of quasiperiodicity in these discrete dynamical systems. AMS classification scheme numbers: 68Q80, 58F07, 70K10, 03D05 PACS numbers: 0270, 6310 peiio:icity Uf i t j