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A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE. PART XIII: BERNOULLI σ_τ-SHIFT RULES

Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA

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and

GIOVANNI E. PAZIENZA

Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA

Cellular Sensory and Wave Computing Laboratory, Computer and Automation Research Institute — MTA-SZTAKI and Pazmany University, Budapest, Hungary

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https://doi.org/10.1142/S0218127410026903Cited by:7 (Source: Crossref)

Abstract

More than one third of the 88 globally-independent Cellular Automata rules exhibit robust simple Bernoulli-shift dynamics. Among them we find rule , which we proved to be chaotic in the previous episodes of our chronicle, and rule , the famous global majority rule. Therefore, we cannot overstate the importance of the Bernoulli σ_τ-shift rules which we will present in two parts of our continuing odyssey on the Nonlinear Dynamics Perspective of Cellular Automata. This paper covers the first 15 of the 30 Bernoulli σ_τ-shift rules. In this paper, after recalling the main concepts of Bernoulli rules — such as the role of the three Bernoulli parameters σ, τ and β — we will display the basin tree diagrams of these rules together with a convenient summary of the results extracted from them. Then, we will show that the superstring is an excellent testing signal to find the robust behavior of a given rule. Finally, we will conclude this paper with a discussion about the difference between robust and nonrobust ω-limit orbits of the Bernoulli σ_τ-shift rules.

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