A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science, Volume 4
Volume IV continues the author's odyssey on l-D cellular automata as chronicled in Volumes I, II and III, by uncovering a novel quasi-ergodicity phenomenon involving orbits meandering among omega-limit orbits of complex (group 5) and hyper (group 6) Bernoulli rules. This discovery is embellished with analytical formulas characterizing the fractal properties of characteristic functions, as well as explicit formulas for generating colorful and pedagogically revealing isomorphic basin tree diagrams. Many new results were derived and proved by uncovering subtle symmetries endowed by various subsets of the 256 Boolean cubes. For the first time, rigorous analyses were used to identify 67, out off 256 , local rules whose asymptotic behaviors consist of robust period-l orbits. The highlight of this continuing odyssey is the discovery of an isolated period-3240 Isle of Eden hidden among the dense omega-limit orbits of Wolfram's remarkable ?random number generating? rule 30. This is the largest gem known to-date and readers are challenged to uncover even larger ones.
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additive rules attractors Isles basin of attraction basin tree diagrams bit string IIIIII bit strings containing Bit-String Code Boolean cubes cellular automata Chua coded by integer color containing exclusively runs converge corresponding deﬁned deﬁnition Distinct Orbit Number dmax Example ﬁnd ﬁnite ﬁring patterns ﬁrst formula Gallery Gardens of Eden ID number IIII IIIII IIIIIIIIII IIIIIIIIII IIIIIIIIII IIIIIIIIIIIIII Isle of Eden isomorphic iterations Kind of Science kmax Lemma llllll maximum transient length n n n n n n n Nonlinear Dynamics Perspective Number of Number Orbit of Length Orbits of Rule patterns of rule period-1 attractor Period-1 Rules period-1 w-limit orbits Period-T coded Period-T Orbit Perspective of Wolfram Proof Quasi-Ergodicity rule 90 runs of k runs of ls scale-free strings containing exclusively strings containing runs Theorem time-1 characteristic function time-1 return map to-limit truth table vertex