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A nonlinear dynamics perspective of Wolfram's new kind of by Leon O Chua

By Leon O Chua

Quantity III keeps the author's quest for constructing a pedagogical, self-contained, but rigorous analytical conception of 1-D mobile automata through a nonlinear dynamics standpoint. utilizing conscientiously conceived and illuminating colour pictures, the worldwide dynamical behaviors of the 50 (out of 256) neighborhood principles that experience no longer but been coated in Volumes I and II are uncovered through their stunningly revealing basin tree diagrams. The Bernoulli -shift dynamics chanced on in quantity II is generalized to carry for all 50 (or 18 globally identical) neighborhood ideas through advanced and hyper Bernoulli wave dynamics. particular international country transition formulation derived for principles 60, ninety, a hundred and five, and one hundred fifty demonstrate a brand new scale-free phenomenon. the main excellent new end result unveiled during this quantity is the Isle of Eden came upon hidden in such a lot (almost 90%) of the 256 neighborhood ideas. Readers are challenged to seek for long-period, remoted Isles of Eden. those are infrequent gemstones ready to be stumbled on.

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List of eight Hyper Bernoulli-shift ΓT N , excluding ΓT N , is called the basin trees of ΓT N . 26 30 41 45 More precisely, 60 106 110 154 An example of a basin tree is shown in Fig. , 2006] for rule 62 with L = 9. In , and this case, ΓT = Γ1 62 = 0♠ ∆ (ΓT ) = B ΓT N Table 13. Steady-state characterization of 88 dynamically-independent local rules. Topologicallydistinct Rules Number Period-1 Rules Period-T Rules T>2 25 13 2 Bernoulli στ -Shift Rules 30 Complex Bernoulli-Shift Rules 10 Hyper Bernoulli-Shift Rules 8 Period-2 Rules Total ∆ =∪ x∈ Observe from Fig.

E. Γn N =Ø Note that our definition of a basin tree ` ´ Γn N . e. bit strings) in the basin of attraction of the ith period-n orbit, where i = 1, 2, . . , m, and m is the total number of period-n orbits. In Gallery 18-1, m = 1 since there is only “one” attractor when L = 3. Hence, i = 1 in Gallery 18-1. In the basin tree Γ1 18 shown in Gallery 18-1, there are all together eight nodes and hence ni = 8. Since L = 3, we have ρ1 = 8/23 = 1. The robustness coefficient ρi in Eq. (16) measures the percentage of initial bit strings which converge to the ith attractor in question.

2. Basin Tree Diagrams of Ten Complex Bernoulli Shift Rules For binary bit strings xn = (xn0 xn1 xn2 ··· xnL−1 ) List of 30 robust Bernoulli στ -shift local rules. 30 Topologically-Distinct Bernoulli στ -shift Rules 2 3 6 7 9 10 11 14 15 24 xn → χ1N (xn ) = xn+1 (3) under local rule N must converge to either a fixed point x∗ = (xn0 ∗ xn1 ∗ xn2 ∗ ··· ∗ xnL−1 ) 42 43 46 56 57 58 74 130 134 138 142 152 162 170 184 ∆ with the length L = I + 1 of the bit strings. , 2006, pp. 1176]. , 2005a], and Tmax = 2L ∆ 25 27 34 35 38 (2) at time n with finite L and periodic (or fixed) boundary conditions, the evolution ∆ Table 9.

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