Narrow Results

We present a cellular-automaton model of a reaction-diffusion excitable system with concentration dependent inhibition of the activator, and study the dynamics of mobile localizations (gliders) and their generators. We analyze a three-state totalistic cellular automaton on a two-dimensional lattice with hexagonal tiling, where each cell connects with 6 others. We show that a set of specific rules support spiral glider-guns (rotating activator-inhibitor spirals emitting mobile localizations) and stationary localizations which destroy or modify gliders, along with a rich diversity of emergent structures with computational properties. We describe how structures are created and annihilated by glider collisions, and begin to explore the necessary processes that generate this kind of complex dynamics.

An interesting question is whether the intrinsic complexity of the gliders in $D$-dimensional cellular automata could be quantitatively analyzed in rigorously mathematical sense. In this paper, by introducing the $D$-dimensional symbolic space, some fundamental dynamical properties of $D$-dimensional shift map are explored in a subtle way. The purpose of this article is to present an accurate characterization of complex symbolic dynamics of gliders in Conway’s game of life. A series of dynamical properties of gliders on their concrete subsystems are investigated by means of the directed graph representation and transition matrix. More specifically, the gliders here are topologically mixing and possess the positive topological entropy on their subsystems. Finally, it is worth mentioning that the method presented in this paper is also applicable to other gliders in different $D$ dimensions.

In the case of one-dimensional cellular automaton (CA), a hybrid CA (HCA) is the member whose evolution of the cells is dependent on nonunique global functions. The HCAs exhibit a wide range of traveling and stationary localizations in their evolution. We focus on HCA with memory (HCAM) because they produce a host of gliders and complicated glider collisions by introducing the hybrid mechanism. In particular, we undertake an exhaustive search of gliders and describe their collisions using quantitative approach in HCAM$(43,74)$. By introducing the symbol vector space and exploiting the mathematical definition of HCAM, we present an analytical method of complex asymptotic dynamics of the gliders.