From: Amara Graps (amara@amara.com)
Date: Sat Jun 08 2002 - 12:52:02 MDT
Hi,
I have converted some old writings in a form to distribute electronically.
Here is a paper I wrote fifteen years ago soon after reading
J. Gleick's Chaos book, to help teach myself the basic ideas.
It's popular science writing, but unpublished. This overview
might be helpful for educators/teachers (freshman university physics
level), or for those of you who are not yet familiar with some of the
physics and math background behind 'emergent order'. I used parts of
this paper some years later in a writing contest to describe using
emergent order to shape a space society.
Enjoy,
Amara
P.S. I'm not particularly interested in paper modifications or debate
unless someone points out a glaring error (in that case I'll withdraw it).
The paper is free: You are welcome to use it as you wish.
Chaos: What is it?
by Amara Graps
1988
Paper: 16 pgs, 13 figs
Formats
PS: 460 kbytes gzipped http://www.amara.com/ftpstuff/Chaos.ps.gz
(1.1Mb unzip)
PDF: 245 kbytes http://www.amara.com/ftpstuff/Chaos.pdf
Contents
Introduction Page 1
Dynamical Systems Page 1
What is an Attractor? Page 3
Strange Attractors and Chaotic Behavior Page 5
The Beauty of Fractals Page 8
Degree of Chaos Page 10
Information Theory Page 10
Emergent Order: A Philosophical Discussion Page 14
Sources Page 15
Introduction
A sector in the scientific community has reshaped the way that we view
the world around us. The short-hand name for it is "chaos." It is a name
for both the discipline itself (also called "nonlinear science,"
"experimental mathematics," and "the study of dynamical systems") and
the physical behavior of a dynamical system when it shows a sensitive
dependence on initial conditions. Chaos is a science of the global
nature of systems. The center of this set of ideas is that simple,
deterministic systems can breed complexity; that systems too complex for
traditional mathematics can yet obey simple laws. Nature is
intrinsically nonlinear and in the past nonlinear was nearly synonymous
with nonsolvable.
An essential difference exists between linear and nonlinear systems.
Mathematically, linear equations are special in that any two solutions
can be added together to form a new solution. As a consequence,
analytic methods have been established for solving any linear system.
You simply break up the complicated system into many simple pieces and
patch together the separated solutions for each piece to form a solution
to the full problem. In contrast, two solutions to a nonlinear system
cannot be added together to form a new solution. The system must be
treated in its full complexity. No general analytic approach exists for
solving them. Nonlinear equations can generate either order or chaos
and for those that generate chaotic motion, there are no useful analytic
solutions.
So what is chaotic motion? And if no useful analytic solutions exist
how can a nonlinear system be modeled? This paper will attempt to
answer those two questions.
--
********************************************************************
Amara Graps, PhD email: amara@amara.com
Computational Physics vita: ftp://ftp.amara.com/pub/resume.txt
Multiplex Answers URL: http://www.amara.com/
********************************************************************
"That's the whole problem with science. You've got a bunch of
empiricists trying to describe things of unimaginable wonder."
--Calvin (& Hobbes)
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