A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART VII: ISLES OF EDEN

Abstract

This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulliστ-shift rules and eight hyper Bernoulliστ-shift rules, the latter including such famous rules [Formula: see text] and [Formula: see text]. All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞. Basin tree diagrams of all ten complex Bernoulli στ-shift rules are exhibited for lengths L = 3, 4, …, 8. Superficial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. Two such properties form the main results of this paper; namely, [Formula: see text] Explicit global state transition formulas are given for local rules [Formula: see text], [Formula: see text] and [Formula: see text]. Such formulas led to the rigorous proof of several surprising periodicity constraints for rule [Formula: see text], and to the discovery of a new global, quasi-equivalence class, defined via an alternating transformation. In particular, local rules [Formula: see text] and [Formula: see text]are globally quasi-equivalent where corresponding space-time patterns can be derived from each other by simply complementing every other row. Another important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules [Formula: see text], [Formula: see text] and [Formula: see text]. In particular, the period "T" of all attractors of rules [Formula: see text], [Formula: see text] and [Formula: see text], as well as of all isles of Eden of rules [Formula: see text] and [Formula: see text], increases linearly with unit slope, in logarithmic scale, with the length L.