Deterministic scale-invariant dynamics in a logistic Game-of-Life model

Scale invariance is a hallmark of criticality in complex dynamical systems. While random external inputs or tunable stochastic interactions are typically required to produce critical behavior, it remains unclear whether scale-invariant dynamics can emerge from purely deterministic interactions. Here, we address this question by studying the asymptotic dynamics of the logistic Game of Life (GOL), a deterministic-parameter extension of Conway’s GOL. In this system, we identify three distinct asymptotic phases separated by two fundamentally different critical points. The first critical point, associated with an unusual form of self-organized criticality, separates a sparse-static phase from a sparse-dynamic phase. The second critical point corresponds to a deterministic percolation transition between the sparse-dynamic phase and a third, dense-dynamic phase. In addition, we observe power-law cluster size distributions with unconventional critical exponents not found in standard equilibrium systems. Overall, our work paves the way for studying emergent scale invariance in purely deterministic systems. Criticality and percolation in dynamical systems are widely studied, yet whether they can emerge from purely deterministic interactions and control parameters remains unclear. Here, the authors reveal deterministic critical points associated with percolation and self-organized criticality in the logistic Game of Life, advancing the understanding of emergent scale invariance in deterministic systems.