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We borrow a working definition for chaos theory from Dr.
Stephen Kellert: The Qualitative Study of Unstable Aperiodic Behavior
in Deterministic Nonlinear dynamical systems. I should briefly
dissect some of these terms to better describe what is and what is
not chaotic in nature:
Chaos is qualitative in that it seeks to know the general
character of a system's long-term behavior, rather than
seeking numerical predictions about a future state. What
characteristics will all solutions of a system exhibit? How
does this system change from exhibiting one behavior to
another?
Chaotic systems are unstable since they tend not to resist
any outside disturbances but instead react in significant
ways. In other words, they do not shrug off external
influences but are partly navigated by them.
The variables describing the state of a system do not
demonstrate a regular repetition of values and are therefore
aperiodic. This unstable aperiodic behavior is highly
complex since it never repeats and continues to show the
effects of the disturbance(s).
These systems are deterministic because they are made up
of few, simple differential equations, and make no
references to implicit chance mechanisms. This is not to be
completely equated with the metaphysical or philosophical
idea of determinism (that human choices could be
predetermined as well).
Finally, a dynamic system is a simplified model for the
time-varying behavior of an actual system. These systems
are described using differential equations specifying the
rates of change for each variable.
Edward Lorenz would stretch the definition of chaos to include
phenomena that are slightly random, provided that their much
greater apparent randomness is not a by-product of their slight true
randomness. In other words, real-world processes that appear to be
behaving randomly - perhaps the falling leaf or the flapping
flag - should be allowed to qualify as chaos, as long as they would
continue to appear random even if any true randomness could
somehow be eliminated.
What this means is when we make slight changes to a system at one
time, and the later behavior of the system may soon become
completely different. In Lorenz' meteorological computer
modeling, he discovered the foundation of mainstream chaos: that
simply-formulated systems with few variables could display highly
complex behavior that was unpredictable and unforseeable. He saw
that slight differences in one variable had profound effects on the
outcome of the whole system. In Chaos parlance, this is referred to
as sensitive dependence on initial conditions. In real weather
situations, this could mean the development of a front or
pressure-system where there never would have been one in
previous models. In differential plotting this took on a new form
called a strange attractor (see figure 1). Initial conditions need not be
the ones that existed when a system was created, but may be the
ones at the beginning of any stretch of time that interests an
investigator.
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