Name
Title | ||
|---|---|---|
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 Bernoulli sigma(tau)-shift rules and eight hyper Bernoulli sigma(tau)-shift rules, the latter including such famous rules [30] and [110]. All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity sigma and Bernoulli return time tau as the length L -> 8.Basin tree diagrams of all ten complex Bernoulli sigma(tau)-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,Rule [90] has no Isles of Eden.Rules [105] and [150] are composed of nothing but Isles of Eden for all string lengths L not divisible by 3.Explicit global state transition formulas are given for local rules [90], [105] and [150]. Such formulas led to the rigorous proof of several surprising periodicity constraints for rule [90], and to the discovery of a new global, quasi-equivalence class, defined via an alternating transformation. In particular, local rules [105] and [150] 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 [90], [105]and [150]. In particular, the period "T" of all attractors of rules [90], [105]and [150], as well as of all isles of Eden of rules [105]and [150], increases linearly with unit slope, in logarithmic scale, with the length L. |
Year | DOI | Venue |
|---|---|---|
2007 | 10.1142/S0218127407019068 | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS |
Keywords | DocType | Volume |
cellular automata, nonlinear dynamics, attractors, Isles of Eden, Bernoulli shift, shift maps, basin tree diagram, Bernoulli velocity, Bernoulli return time, complex Bernoulli shifts, hyper Bernoulli shifts, rule 90, rule 105, rule 150, binomial series, scale-free phenomena | Journal | 17 |
Issue | ISSN | Citations |
9 | 0218-1274 | 14 |
PageRank | References | Authors |
1.06 | 8 | 5 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Leon O. Chua | 1 | 1860 | 497.65 |
| Junbiao Guan | 2 | 34 | 7.69 |
| Valery I. Sbitnev | 3 | 167 | 25.47 |
| Jinwook Shin | 4 | 51 | 6.27 |
| Jinwook Shin | 5 | 51 | 6.27 |