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

By Leon O. Chua

This novel ebook introduces mobile automata from a rigorous nonlinear dynamics point of view. It offers the lacking hyperlink among nonlinear differential and distinction equations to discrete symbolic research. an incredibly important interpretations of mobile automata by way of neural networks can also be given. The ebook offers a scientifically sound and unique research, and classifications of the empirical effects awarded in Wolfram s huge New form of Science.
Volume 2: From Bernoulli Shift to 1/f Spectrum; Fractals in every single place; From Time-Reversible Attractors to the Arrow of Time; Mathematical starting place of Bernoulli -Shift Maps; The Arrow of Time; Concluding comments.

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Additional resources for A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science, Volume 2

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Only one out of many attractors is shown for each time-1 map in Fig. 10. In Fig. 10(a), only one period-1 attractor of rule 200 is shown (labeled as point 1 ). The domain of the time-1 map ρ1 [200] in this trivial case consists of only the single point { 1 }, and all iterates 383 map trivially onto the fixed point 1 . One can interpret point 1 as the point where a planet intersects an imaginary Poincare cross-section once every revolution. Figure 10(b) shows a period-2 attractor (out of many others) of local rule 51 .

All other period-2 points form clusters and are painted in dull blue color. The power spectrum of all period-2 time-1 maps consists of a bold red line located at f = 1/2. The power spectrum of the corresponding backward time-1 map is qualitatively identical and is therefore redundant. 388 A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 2. Gallery of forward time-1 maps ρ1 [N ] and backward time-1 maps ρ†1 [N ] for attractor Λ1 (red), Λ2 (blue), and Λ3 (green). Chapter 4: From Bernoulli Shift to 1/F Spectrum Table 2.

In other words, the CA characteristic function χ1N is a complete and global representation of N . It is complete because it contains all information needed to derive the dynamic evolutions from any initial state by simply drawing a Lameray (cobweb) diagram directly on χ1N ! It is global because each point on χ1N codes for an entire configuration of length I + 1, and not just for one pixel if the local rule were used instead. Clearly, the points defining the time-1 maps ρ1 [N ] and ρ†1 [N ] are subsets of the points defining the characteristic function χ1N .

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