Chaotic Dynamics and Fractals by Michael F. Barnsley, Stephen G. Demko

By Michael F. Barnsley, Stephen G. Demko

1986 1st Ed. Vol. 2 educational Press

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But the (hopefully) succinct description will compress the perceived information concerning the flower to something equivalent to part of what is stored in the seed, will be convenient to handle, and may be the basis of discovering new empirical laws; just as Kepler by describing planetary motion well in terms of ellipses gave Newton the neatly compressed information 56 he needed. 2. THE COLLAGE THEOREM The following is a simplified discussion of material in [1, 2, 7, 9 ] . To approach the inverse problem we begin with the chaos game (which I have taught twelve year old children to play with interest).

By confining its attention to that which is possible, complexity theory has rendered all dynamics susceptible to meaningful approximation. 4. ALGORITHMIC RANDOMNESS In this section, we at last directly confront the impos­ sible, seeking to solve the unsolvable, predict the unpredic­ table. In particular, we here show by example how one may analyze chaotic systems for which all or nearly all orbits have positive complexity. In order to physically motivate our analysis, let us very briefly review the long-standing search for a first principles derivation of diffusive energy transport (heat flow) in electrically insulating solids.

Ductivity value stabilizes above N = l l . The con­ This figure clearly illustrates the transition of energy transport from sound to heat. Figure 10 also emphasizes that randomness in the heat reservoirs is not the source of diffusive energy transport. Having validated the Fourier heat law using thermal reservoir, steady state calculations, we next sought an inde­ pendent verification using the Green-Kubo linear transport theory formula for the conductivity which, very loosely speaking, derives from the diffusive spread of a localized energy packet in an isolated system.

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