Step Size is a Consequential Parameter in Continuous Cellular Automata

Abstract: Step size in continuous cellular automata (CA) plays an important role in the stability and behavior of self-organizing patterns. Continous CA dynamics are defined by formula very similar to numerical estimation of physics-based ordinary differential equations, specifically Euler's method, for which a large step size is often inaccurate and unstable. Rather than asymptotically approaching more accurate estimates of CA dynamics with decreasing step size, continuous CA may support different self-organizing patterns at different ranges of step size. We discuss several examples of mobile patterns that become unstable at step sizes that are too small as well as too large. Additionally, an individual mobile pattern may exhibit qualitatively different behavior across a range of step sizes. We demonstrate examples of the effects of step size in pattern stability and qualitative behavior in continuous CA implemented in the Lenia framework and its variant, Glaberish.