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 standpoint. It provides the lacking hyperlink among nonlinear differential and distinction equations to discrete symbolic research. a shockingly worthwhile interpretations of mobile automata when it comes to neural networks is additionally given. The publication offers a scientifically sound and unique research, and classifications of the empirical effects offered in Wolfram s enormous New type 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 info for A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science, Volume 2

Example text

2003], it follows that all orbits of 15 , 51 , 85 , 204 , and 240 are invariant as well. It has been verified by exhaustive computer simulation that only the six rules belonging to S41 are endowed with only invariant orbits. In general, invariant orbits have noninvariant neighboring orbits. We have seen earlier a special case of an invariant orbit consisting of only a single point; namely, φ 128 = 1• 00 in Fig. 3(a). Observe that in addition to having no basin of attraction, φ 128 = 1• 00 has no preimage (predecessor).

1. Mapping CA attractors onto time-τ maps Since invariant orbits are not attractors, they are not robust in the sense that precisely specified initial states must be used to observe them. Since one of the most fundamental problems in nonlinear dynamics is to analyze and predict their longterm behaviors as t → ∞, we will develop some novel and effective techniques for analyzing and predicting global qualitative behaviors of robust CA attractors. In general, each CA local rule N can exhibit many distinct attractors Λi , i = 1, 2, .

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.

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