Cellular Automata, Dynamical Systems and Neural Networks by François Blanchard (auth.), Eric Goles, Servet Martínez

By François Blanchard (auth.), Eric Goles, Servet Martínez (eds.)

This e-book comprises the classes given on the 3rd tuition on Statistical Physics and Cooperative platforms held at Santiago, Chile, from 14th to 18th December 1992. the most concept of this periodic university used to be to assemble scientists paintings­ with fresh traits in Statistical Physics. extra accurately ing on topics comparable comparable with non linear phenomena, dynamical structures, ergodic idea, mobile au­ tomata, symbolic dynamics, huge deviation idea and neural networks. Scientists operating in those topics come from a number of parts: arithmetic, biology, physics, computing device technological know-how, electric engineering and synthetic intelligence. lately, an important cross-fertilization has taken position with reference to the aforesaid medical and technological disciplines, so that it will provide a brand new method of the learn whose universal center continues to be in statistical physics. every one contribution is dedicated to at least one or extra of the former topics. normally they're dependent as surveys, providing whilst an unique perspective in regards to the subject and displaying as a rule new effects. The expository textual content of Fran

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X2r+I) ,X2r+l 2:::: Xj+12 2 r-j. j=O Different rules may have the same attractor (Fig. 1b). Due to different time correlations, their spatiotemporal patterns are, however, different. 31 sequence of zeros of even length, configurations generated by Rule 18 may contain defects of one type only. 1a). This process has been studied by Grassberger (1983) who found that, starting from a random initial configuration, the density of defects decreases as c 1 12 . 10 These defects may be viewed as particles.

The evolution rule is synchronous, that is, all sites are updated simultaneously. CAs are, therefore, discrete (in space and time) dynamical systems. They may be more precisely defined as follows. Let s: Z x N t--? {0, 1} be a function that satisfies the equation (Vi E Z) (Vt E N) s(i,t + 1) = f(s(i- r,t),s(i- r + 1,t), ... ,s(i + r,t)) and such that (Vi E Z) s( i, 0) = so( i), 29 where N is the set of nonnegative integers, Z the set of all integers, and Z --t {0, 1} a given function that specifies the initial condition.

Introduction The first task that faces the theoretician who wants to interpret the time evolution of a complex system is the construction of a model. In the actual system many features are likely to be important. Not all of them, however, should be included in the model. Only the few relevant features which are thought to play an essential role in the interpretation of the observed phenomena should be retained. Such simplified descriptions should not be criticized on the basis of their omissions and oversimplifications.

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