From deterministic cellular automata to coupled map lattices

Abstract

A general mathematical method is presented for the systematic construction of coupled map lattices (CMLs) out of deterministic cellular automata (CAs). The entire CA rule space is addressed by means of a universal map for CAs that we have recently derived and that is not dependent on any freely adjustable parameters. The CMLs thus constructed are termed real-valued deterministic cellular automata (RDCA) and encompass all deterministic CAs in rule space in the asymptotic limit $\kappa \to 0$ of a continuous parameter $\kappa$. Thus, RDCAs generalize CAs in such a way that they constitute CMLs when $\kappa$ is finite and nonvanishing. In the limit $\kappa \to \infty$ all RDCAs are shown to exhibit a global homogeneous fixed-point that attracts all initial conditions. A new bifurcation is discovered for RDCAs and its location is exactly determined from the linear stability analysis of the global quiescent state. In this bifurcation, fuzziness gradually begins to intrude in a purely deterministic CA-like dynamics. The mathematical method presented allows to get insight in some highly nontrivial behavior found after the bifurcation.

arXiv comment19 pages, 9 figures, 60 references. Paragraphs added to introduction and conclusions. Some new references added. Accepted to J. Phys. A: Math. Theor

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Cited by in corpus (7)

• A Comprehensive Taxonomy of Cellular Automata
• Weakly coupled map lattice models for multicellular patterning and collective normalization of abnormal single-cell states
• Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding
• Unifying vectors and matrices of different dimensions through nonlinear embeddings
• On the nontrivial zeros of the Dirichlet eta function
• A constructive theory of shape
• On the characteristic function of a collection of sets