By Leon O. Chua
Quantity IV maintains the author's odyssey on l-D mobile automata as chronicled in Volumes I, II and III, via uncovering a singular quasi-ergodicity phenomenon related to orbits meandering between omega-limit orbits of complicated (group five) and hyper (group 6) Bernoulli principles. This discovery is decorated 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 through uncovering sophisticated symmetries endowed through a number 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 include powerful 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 notable "random quantity producing" rule 30. this is often the most important gem identified to-date and readers are challenged to discover even higher ones.
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Extra info for A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science, Vol. 4
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A cursory inspection of the 256 Boolean cubes listed in Table 1 would extract, in a few minutes, 104 local rules with a complexity index κ = 1, namely, those Boolean cubes whose red vertices can be separated from the blue vertices by no more than This list is not unique in the sense that one can pick many other groups containing 88 independent rules. Our choice is obtained by scanning the 256 rules from N = 0 to N = 255 , and deleting any rule that is equivalent to a previously listed rule. , 2007a, 2007b].
The φn — versus — φn−1 return map of this period-3240 Isle of Eden is shown in Fig. 10(a). Note the very messy plot gives very little information. However, if we plot this Isle of Eden orbit as a τ = 480 return map in Fig. 10(b), we see the φn — versus — φn−τ time-τ map is virtually identical to the Bernoulli mapD φn = 2φn−τ mod 1 (14) associated with the time-1 return map of rule 170 . Since it is well known that the Bernoulli map Eq. (14) is ergodic, Fig. 1 of Sec. 1. Fig. 9. Orbit of a period-3240 (with σ = 1, τ = 480) Isle of Eden of 30 with L = 27.