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 snap shots, the worldwide dynamical behaviors of the 50 (out of 256) neighborhood ideas that experience now not but been lined 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 similar) neighborhood principles through advanced and hyper Bernoulli wave dynamics. particular international kingdom transition formulation derived for ideas 60, ninety, a hundred and five, and one hundred fifty show a brand new scale-free phenomenon. the main brilliant new consequence unveiled during this quantity is the Isle of Eden discovered 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 gem stones ready to be came upon.

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6(a)-i. The basin tree of Γ3 62 of bit strings (10) is the set ∆ , 23♠ , 60♠ , 1♠ , 35♠ , 22♠ (Γ3 ) = 40♠ (11) In this case, one can associate the basin tree (Γ3 ) ♠ ♠ , 60♠ , 1♠ , 35 , 22 } as two subtrees 40♠ and {23♠ emerging from the period-3 orbit Γ3 62 , which is analogous to a cluster of roots. For large L, a basin tree in general is made of many topologically similar subtrees, such as Fig. , 2006]. In this case, we have a period-14 orbit ✓✏ ✓✏ ♠ ♠ ♠ ♠ , 55♠ , Γ( 62 ) = 59 , 102 , 93 , 51 , 110 , 89 ✒✑ ✓✏ ✒✑ ✓✏ ✓✏ ♠ ♠ 108 , 91 , 54♠ , 109 , 27♠ , 118 , 77 ✒✑ ✒✑ ✒✑ (12) and the basin tree Γ14 62 of Γ14 62 is made of seven subtrees having identical topologies.

The dynamics on each attractor is a Bernoulli στ -shift with σ1 = 5, τ = 2, or σ2 = −5, τ = 2, as depicted in the φn−2 → φn time-2 map. The time-2 map φn−2 → φn consists of β = 2σ1 = 32 parallel red Bernoulli lines with slope 2σ1 = 32, or equivalently, to β = 2|−σ2 | = 32 parallel blue Bernoulli lines with slope 2σ2 = 1/32. Observe that the two red dots now fall on the diagonal of the time-2 map, as expected of period-2 orbits. Again, β > 0 because the slope of each red (or blue) Bernoulli line is positive.

May 6, 2009 28 10:6 ch01 A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Also listed on top of each gallery is the robustness coefficient and 1•2 +1•2 +0•2 =6 2 1 0 respectively. These numbers are enclosed by small circles, and are represented as nodes of a digraph where a directed edge pointing from node S♠ 1 to ♠ node S2 means that bit string S1 maps to bit string S2 after one iteration under rule N . For example, Gallery 18-1 shows the basin trees Γ1 18 = 7♠ ; 2♠ 5♠ ; 4♠ , 3♠ ; 1♠ , 6♠ converging to a period-1 (fixed point) orbit Γ1 18 = { 0 }.

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