By Leon O. Chua
Quantity IV keeps the author's odyssey on l-D mobile automata as chronicled in Volumes I, II and III, through uncovering a unique quasi-ergodicity phenomenon concerning orbits meandering between omega-limit orbits of complicated (group five) and hyper (group 6) Bernoulli ideas. This discovery is adorned with analytical formulation characterizing the fractal houses of attribute services, in addition to particular formulation for producing colourful and pedagogically revealing isomorphic basin tree diagrams. Many new effects have been derived and proved by means of uncovering refined symmetries endowed via numerous subsets of the 256 Boolean cubes. For the 1st time, rigorous analyses have been used to spot sixty seven, out off 256 , neighborhood ideas whose asymptotic behaviors include strong period-l orbits. The spotlight of this carrying on with odyssey is the invention of an remoted period-3240 Isle of Eden hidden one of the dense omega-limit orbits of Wolfram's impressive "random quantity producing" rule 30. this can be the most important gem identified to-date and readers are challenged to discover even higher ones.
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Additional resources for A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science, Vol. 4
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List of 104 Linearly-Separable Boolean Function Rules. 0 2 1 1 15 17 16 4 48 80 84 81 4 128 138 136 1 176 4 213 4 241 243 4 Color Legend 4 221 1 1 244 4 245 4 Period-1 N Rule 224 4 1 To derive the left-right transformation N † (N ) of any local rule N , simply obtain the 0 mirror0image of the 0 Boolean0cube N about the main diagonal plane (shown shaded in Fig. 7(a)) pass. Note that this operaing through the vertex 0♠ tion can be implemented by identifying each pair of symmetrically located vertices with opposite colors, and then complementing the colors.
The union of all these local rules gives only 28 distinct Permutive rules, as exhibited in Table 11. We will show in the following sections that Permutive rules possess some remarkable properties. D depicted in Table 13. It is easy to prove that each rule N in Table 13 has a unique decomposition via the eight Boolean cubes basis functions in Table 12. 5. Superposition of local rules The eight Boolean cubes exhibited in Table 12 are independent in the sense that it is impossible to decompose any of them into the “union” of two or more simpler Boolean cubes by taking the logic “OR” operation between the colors of corresponding vertices, where “red ” is coded “1” and “blue” is coded “0”, respectively.
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science, Vol. 4 by Leon O. Chua