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 study establishes that certain nonlinear automata on finite lattices may be mapped exactly onto a linear automation, thus providing an 'exact solution' for the nonlinear systems, and permitting description of their fundamental dynamical features such as limit cycle period, attractor structure, and transience length. These particular nonlinear automata generate multiple domains within which evolution exactly mimics that of the linear automation, with the domain wall behaviour itself governed by the dynamics of the linear automation. 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 periodicity of its spatial behaviour. The limit cycle behaviour of the nonlinear automata 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 be viewed as providing a realization of quasiperiodicity in these discrete dynamical systems.