[Paper Review] Introducing Lyapunov profiles of cellular automata

Jan Baetens, Janko Gravner|arXiv (Cornell University)|Sep 22, 2015

Cellular Automata and Applications8 references3 citations

TL;DR

This paper introduces Lyapunov profiles as a unified framework for analyzing cellular automata (CAs), combining the directional spread of perturbations (defect cone widening) and defect accumulation intensity into a single spatially resolved measure. The key contribution is a vector-based Lyapunov profile that captures both the speed and intensity of defect propagation across each cell, revealing distinct dynamical behaviors across Wolfram's Class 3 and Class 4 rules, including asymmetric propagation and phase transitions in defect dynamics.

ABSTRACT

In line with the stability theory of continuous dynamical systems, Lyapunov exponents of cellular automata (CAs) have been conceived two decades ago to quantify to what extent their dynamics changes following a perturbation of their initial configuration. More precisely, Lyapunov exponents of CAs have either been understood as the rate by which the resulting defect cone widens as these dynamical systems are evolved from a perturbed initial configuration, or as the rate by which defects accumulate during their evolution. The former viewpoint yields insight into the extent of the affected region, whereas the latter tells us something about the intensity of the defect propagation. In this paper, we will show how these viewpoints can be united by relying on Lyapunov profiles of CAs.

Motivation & Objective

• To unify two existing viewpoints on Lyapunov exponents in cellular automata: defect cone widening and defect accumulation intensity.
• To develop a spatially resolved measure that captures both the direction and rate of defect propagation across individual cells.
• To provide a comprehensive stability analysis tool for cellular automata by replacing scalar exponents with a vector-based profile.
• To characterize the dynamical behavior of elementary cellular automata (ECAs), particularly in Wolfram's Classes 3 and 4, using this new profile framework.
• To explore the implications of Lyapunov profiles for understanding defect propagation, localization, and phase transitions in CA dynamics.

Proposed method

• Define a defect vector ε̃_t where ε̃_t^i is the number of defects at cell i at time t, tracking both position and multiplicity of perturbations.
• Compute the time-averaged rate of defect accumulation per cell using λ̃_T^i = (1/T) ∑_{t=1}^T (1/t) log(ε̃_t^i / ε̃_0^i), forming the Lyapunov profile.
• Normalize the profile to allow comparison across different rules and system sizes, enabling visualization of spatial dynamics.
• Evolve both the original and perturbed initial configurations to compute defect propagation over time, preserving spatial information.
• Use heat maps to visualize the time evolution of the normalized Lyapunov profile for dynamic analysis of defect spread.
• Classify rules based on profile shape—e.g., symmetric, asymmetric, discontinuous, or smooth—linking structure to dynamical behavior.

Experimental results

Research questions

• RQ1 How can the two distinct interpretations of Lyapunov exponents in CAs—defect cone widening and defect accumulation intensity—be unified into a single analytical framework?
• RQ2 What structural and dynamical features of cellular automata are revealed by spatially resolved Lyapunov profiles that scalar exponents cannot capture?
• RQ3 How do Lyapunov profiles differ across Wolfram’s Classes 3 and 4 elementary cellular automata in terms of symmetry, phase transitions, and propagation speed?
• RQ4 To what extent do Lyapunov profiles reveal non-trivial defect propagation in Class 2 rules, including localized or confined dynamics?
• RQ5 Where is the maximum of the Lyapunov profile located relative to the initial defect, and what does this imply about the directionality of defect propagation?

Key findings

• Rule 30 exhibits a highly asymmetric Lyapunov profile with rapid rightward defect propagation (maximum speed) and slower leftward spread, peaking at i ≈ -2300, indicating a sharp transition from expansive to unaffected cells.
• Rule 150 shows symmetric defect cone widening at maximum speed but with lower defect accumulation intensity compared to rule 30.
• Class 4 rules such as 106 and 110 display multiple phase transitions, asymmetry, and non-smooth profiles, with the maximum Lyapunov value not located at the initially perturbed cell.
• Rule 57 is the only Class 2 rule that produces a Lyapunov profile spanning the entire light cone [-T, T], indicating widespread but non-localized defect propagation.
• Rule 62 exhibits a unique profile with rightward propagation at maximum speed and inward-pushing left boundary, confirmed by a time-evolving heat map showing confined, dynamic defect regions.
• Discontinuous profiles in Class 2 rules (e.g., 28, 33, 37, 73, 108, 156) indicate localized defect accumulation with no long-range propagation, consistent with λ̃_T^i = -∞ in most cells.