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On the topological sensitivity of cellular automata
Jan M Baetens1, Bernard De Baets
1KERMIT, Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, Gent, Belgium. Jan.Baetens@UGent.be
Chaos (Woodbury, N.Y.)
|July 5, 2011
Summary
This study introduces topological Lyapunov exponents to quantify how cellular automata (CA) dynamics change with topology. This new method addresses a gap in understanding CA topological sensitivity.
Area of Science:
- Complex systems
- Computational mathematics
- Discrete dynamical systems
Background:
- Cellular automata (CA) are discrete dynamical systems studied for their sensitivity to initial conditions.
- The impact of topology on CA dynamics has been largely overlooked, lacking quantitative measures.
Purpose of the Study:
- To introduce and validate methods for quantifying the topological sensitivity of cellular automata.
- To bridge the gap in understanding the relationship between CA topology and their emergent dynamics.
Main Methods:
- Proposed topological Lyapunov exponents to measure trajectory divergence under topological perturbations.
- Introduced topological derivatives as a measure of CA topological sensitivity.
- Applied and validated the methodology across 256 elementary CA and irregular totalistic CA.
Main Results:
- Demonstrated the effectiveness of topological Lyapunov exponents in characterizing CA topological sensitivity.
- Provided a quantitative framework for analyzing the influence of topology on CA behavior.
- Successfully applied the novel methods to diverse CA models.
Conclusions:
- The proposed topological Lyapunov exponents and derivatives offer a robust way to measure CA topological sensitivity.
- This work opens new avenues for exploring the interplay between spatial structure and dynamics in cellular automata.
- Highlights the critical, yet understudied, role of topology in CA behavior.
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