Summary: 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 structure, and transience length. These particular nonlinear automata generate multiple domains within which evolution exactly mimics that of the linear automaton, with the domain wall behaviour itself governed by the dynamics of the linear automaton. 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 viewed as providing a realization of quasiperiodicity in these discrete dynamical systems.