nemo@rochester.UUCP (Wolfe) (10/07/85)
[lost your appetite?]
I recently read an article from the NY Times magazine that introduced
me to a field of study in Physics called "chaos". The basic idea was
that physicists could get a handle on choatic phenomena (like water
going over a waterfall, or vortices, or cigarette smoke) and particularly
the critical transition from orderly to chaotic (ie: where the smoke
starts curling crazily) by the character of some equations. These
were used iteratively, plugging the new iterate into the equation to
get the next one. In general, the equations (meaning what happens to
the sequence) are very sensitive to starting conditions, which is why
physicists have had a rough time with non-linear, chaotic events. The
reason I bring this up is the similarity with both computing fractals
and their appearance. One way I have heard them described is in terms
of the way the pattern repeats itself on all levels of magnification.
The same is true for these beasties being played with in chaos. Does
anyone know more about this field (chaos) or the relationship between
the mathematics they study and fractals? It seems that the hope I
heard expressed that fractals could open up understanding of many hitherto
dficult fields (like metal crystalization, etc) is already a fact (by
chaos in economics, fluid flow, laser irregularities, etc).
Nemo
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Rochester, NY 14627gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (10/08/85)
This is the field of "dynamical systems", which has recently undergone a large amount of active development by a number of mathematicians. There is indeed a close relationship among "chaos", fractals, iteration theory, and dynamical systems. I am not sure that the theory helps much in predicting viscosity of turbulent flow, etc. but it might. There is a series of good books with lots of nice diagrams targeted at about the undergraduate level; I left mine at home but I think Abraham was one of the authors. Someone else will probably post the accurate title, author, etc. but if they don't I'll try to do it later. Back when I was working in solid-state theory, iterative "renormalization group" approaches were leading to some interesting discoveries about critical-point phenomena. I don't know whether the theory of dynamical systems has supplanted this line of inquiry or not.
jbuck@epicen.UUCP (Joe Buck) (10/12/85)
In article <12113@rochester.UUCP> nemo@rochester.UUCP (Wolfe) writes: >... Does >anyone know more about this field (chaos) or the relationship between >the mathematics they study and fractals? It seems that the hope I >heard expressed that fractals could open up understanding of many hitherto >dficult fields (like metal crystalization, etc) is already a fact (by >chaos in economics, fluid flow, laser irregularities, etc). >Nemo That's exactly what Benoit Mandelbrot, the man who coined the term "fractal", has tried to do. I recommend reading his books "Fractals: Form, Chance, and Dimension" and "The Fractal Geometry of Nature". He shows how fractals may apply not only to the shape of mountains, continents, etc. but to the distribution of stars and galaxies, to fluid turbulence, the distribution of errors on a bursty channel, the behavior of a fluid near the critical point, and more, though he just touches on each field. -- Joe Buck | Entropic Processing, Inc. UUCP: {ucbvax,ihnp4}!dual!epicen!jbuck | 10011 N. Foothill Blvd. ARPA: dual!epicen!jbuck@BERKELEY.ARPA | Cupertino, CA 95014
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (10/14/85)
The books I mentioned are part of "The Visual Mathematics Library": Vol.0 "Manifolds and Mappings" "Dynamics -- The Geometry of Behavior" Vol.1 "Part One: Periodic Behavior" ISBN 0-942344-01-4 Vol.2 "Part Two: Chaotic Behavior" ISBN 0-942344-02-2 Vol.3 "Part Three: Global Behavior" Vol.4 "Part Four: Bifurcation Behavior" by Ralph H. Abraham & Christopher D. Shaw Aerial Press, Inc., P.O.Box 1360, Santa Cruz, CA 95061 Volumes 1 & 2 are available; I have been informed that Volume 3 is about ready. There are plans to produce films and floppy disks of the phase portraits; I don't know if any are available yet.