Using Dynamical Systems Theory to Quantify Complexity in Asymptotic Lenia

TL;DR

This paper applies dynamical systems theory to continuous cellular automata, specifically Asymptotic Lenia, to rigorously analyze and quantify emergent complexity through mathematical tools like attractors, Lyapunov exponents, and fractal dimensions.

Contribution

It introduces a mathematical framework for analyzing complexity in CCAs using PDE formulations, including conditions for attractors and effective complexity measures, with an open-source implementation.

Findings

Identification of four solution classes in Asymptotic Lenia

Conditions for the existence of a global attractor with fractal dimension >4

Kaplan-Yorke dimension as a complexity measure

Abstract

Continuous cellular automata (CCAs) have evolved from discrete lookup tables to continuous partial differential equation (PDE) formulations in the search for novel forms of complexity. Despite innovations in qualitative behavior, analytical methods have lagged behind, reinforcing the notion that emergent complexity defies simple explanation. In this paper, we demonstrate that the PDE formulation of Asymptotic Lenia enables rigorous analysis using dynamical systems theory. We apply the concepts of symmetries, attractors, Lyapunov exponents, and fractal dimensions to characterize complex behaviors mathematically. Our contributions include: (1) a mathematical explanation for the four distinct solution classes (solitons, rotators, periodic and chaotic patterns), (2) conditions for the existence of a global attractor with fractal dimension $>4$, (3) identification of Kaplan-Yorke dimension as an effective complexity measure for CCAs, and (4) an efficient open-source implementation for calculating Lyapunov exponents and the covariant Lyapunov vectors for CCAs. We conclude by identifying the minimal set of properties that enable complex behavior in a broader class of CCAs. This framework provides a foundation for understanding and measuring complexity in artificial life systems.