Step Size is a Consequential Parameter in Continuous Cellular Automata

TL;DR

This paper investigates how the step size parameter critically influences the stability and behavior of patterns in continuous cellular automata, revealing that different step sizes can lead to distinct self-organizing phenomena.

Contribution

It demonstrates that step size in continuous CA significantly affects pattern stability and behavior, challenging the assumption that smaller step sizes always yield more accurate results.

Findings

Large step sizes cause instability in patterns.

Different step sizes lead to qualitatively different behaviors.

Pattern stability varies non-monotonically with step size.

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.