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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of by Leon O. Chua

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 principles. This discovery is adorned with analytical formulation characterizing the fractal houses of attribute capabilities, in addition to specific formulation for producing colourful and pedagogically revealing isomorphic basin tree diagrams. Many new effects have been derived and proved via uncovering sophisticated symmetries endowed via a variety of subsets of the 256 Boolean cubes. For the 1st time, rigorous analyses have been used to spot sixty seven, out off 256 , neighborhood principles whose asymptotic behaviors encompass 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 awesome "random quantity producing" rule 30. this is often the most important gem recognized to-date and readers are challenged to discover even better ones.

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Additional resources for A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science, Vol. 4

Example text

8 There are some rules, however, where the local complementation T c (N ) coincides with the global complementation T (N ) to be defined below. In such cases, the space-time patterns of N and N c are also complements of each other for all t. 7 February 14, 2011 15:12 ch01 D Chapter 1: Quasi-Ergodicity 33 Table 5. A gallery of 104 linearly-separable local rules. The red color engulfing each rule number N implies a complexity index κ = 1 for all 104 rules. : List of 104 Linearly-Separable Boolean Function Rules.

Chapter 1: Quasi-Ergodicity • •• ~t--~. • : L-Permutive ••• ••• ... ••• R-Permutive . oIt--...... 1~~ ... --"'--e. : R-Permutive ... ••• • •• ... H ... --"'--e. %11: B-Permutive 1mI: R-Permutive 1m: R-Permutive 1m: B-Permutive ... ~~~...... ••• . . ---t..... ••• ••• ... ••• ... ••• 111m: R-Permutive . : L-Permutive 1m: L-Permutive ... ... :.... ~t-~ ~~..... ••• ... ••• ... -~~. IIP): R-Permutive ••• ... ••• ... w~~~ ... --"'--e. IIIJ: B-Permutive 1m: R-Permutive 1m: R-Permutive 1m: B-Permutive ..

L-Permutive • •• • •• II"t-+---1~. DDJ: ... --~. L-Permutive 43 44 4 6 4 6 5 16 0 7 1 4 6 5 7 5 7 32 0 2 3 0 2 4 6 2 5 1 2 1 0 7 3 1 3 1 3 4 6 4 6 5 7 5 7 64 0 2 4 0 2 Eight Boolean cube basis functions. 1 3 1 3 4 6 4 6 5 7 5 7 128 0 2 8 0 2 1 3 1 3 15:12 2 Table 12. February 14, 2011 ch01 45 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 4 2 2 4 4 4 2 2 2 4 18 18 18 18 18 18 18 18 8 2 31 16 16 2 43 1 2 42 8 8 8 41 16 1 8 40 4 16 2 1 39 16 4 4 37 1 4 36 2 35 1 2 34 32 32 32 32 32 32 32 32 32 32 32 53 16 65 64 63 62 61 60 59 58 57 56 55 54 52 51 16 33 8 8 32 4 4 16 50 8 49 16 4 48 16 16 47 16 8 46 16 4 8 8 45 16 64 128 44 32 16 16 32 1 2 1 1 30 29 28 2 27 1 2 26 8 1 25 8 8 4 4 4 24 2 23 1 2 2 22 1 2 64 128 38 32 16 16 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 into superposition of eight basis characteristic functions.

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