By Leon O. Chua
This novel ebook introduces mobile automata from a rigorous nonlinear dynamics viewpoint. It offers the lacking hyperlink among nonlinear differential and distinction equations to discrete symbolic research. an incredibly helpful interpretations of mobile automata by way of neural networks can be given. The ebook offers a scientifically sound and unique research, and classifications of the empirical effects offered in Wolfram s huge New type of Science.
Volume 2: From Bernoulli Shift to 1/f Spectrum; Fractals in all places; From Time-Reversible Attractors to the Arrow of Time; Mathematical beginning of Bernoulli -Shift Maps; The Arrow of Time; Concluding feedback.
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Additional info for A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science, Volume 2
Example 3. χ1170 χ1 The graph of the characteristic function 170 of 170 is shown in Fig. 4(a). Note that there are no period-1 ﬁxed points except at φ 170 = 0• 00 and φ 170 = 1• 00. Observe also the vertices of all vertical lines fall on one of two parallel lines with slope = 2. This is an example, par excellence, of the classic Bernoulli shift [Nagashima & Baba, 1999], a subject to be discussed at length in Sec. 5. Example 4. 379 blue vertical lines terminate on the upper parallel straight lines.
162 and χ362 The graphs of the “time-1” characteristic function χ162 and “time-3 ” characteristic function χ362 of 62 are shown in Figs. 7(a) and 7(b), respectively. Observe that while there are no period-1 ﬁxed points in χ162 , there are many vertical lines which landed on the main diagonal of χ362 . This implies that 62 has many period-3 attractors. Such local rules will be studied in Sec. 3. χ1240 The graph of the characteristic function χ1240 of 240 is shown in Fig. 4(b). 2 The “double-valued” appearance is only illusory because all red vertical lines terminate on the lower straight lines of slope = 1/2, and all 3.
Observe that no garden of Eden can be a periodic orbit with a period TΛ > 1, otherwise any point on the orbit is a predecessor of its next iterate. A period-1 garden of Eden is therefore a truly unique specie worthy of its own name, henceforth dubbed an isle of Eden. Indeed, we can generalize this unique phenomenon, which does not exist in continuous dynamical systems (such as ODE), to deﬁne a “period-k ” isle of Eden from the kth iterated characteristic function χkN of N . A gallery of period-k isles of Eden of all one-dimensional cellular automata will be presented in Part V of this tutorial series.