Dynamics in a Low-Rank Separable Field Cellular Automaton

Abstract

Cellular automata (CA) are widely used to study how simple rules generate life-like behavior, typically through local interactions. Here, we introduce a Separable-Field Cellular Automaton (SFCA), in which conventional local-neighborhood interactions are replaced by updates governed by a normalized global coupling field. Each cell is updated according to q = n/n_max, with survival and birth controlled by threshold intervals S and B. Systematic scans over interval widths and positions revealed four outcome classes: extinction, fixed points, cycles, and long transients. The phase diagram was organized by the relative geometry of S and B: fixed points were enriched when B was contained within S, whereas long transients concentrated near the boundary between partial overlap and no overlap. A fine scan across this transition revealed a narrow long-transient ridge separating two distinct cycle-dominated regimes: a dense, high-change-rate branch approximating global period-2 alternation and a sparse, low-change-rate, stripe-like branch. Damage-spreading analyses were consistent with delayed selection between these cyclic attractor families and provided no evidence for sustained exponential trajectory separation. Finite-size analyses showed that the ridge persists across all tested lattice sizes, although its position shifts systematically with system size. These results show that structured long-lived dynamics can arise under separable global coupling with a field matrix of rank at most one, suggesting that nontrivial collective organization can persist under strongly compressed interaction architectures.