Chaos Overview
What is chaos theory?
Formally, chaos theory is defined as the study of
complex nonlinear dynamic systems. Complex implies just
that, nonlinear
implies recursion and higher mathematical algorithms, and dynamic
implies nonconstant and nonperiodic. Thus chaos theory is, very
generally,
the study of forever changing complex systems based on mathematical
concepts
of recursion, whether in the form of a recursive process or a set of
differential
equations modeling a physical system.
For a more rigorous definition of chaos theory,
it is
advisable to visit the much more scientific, much more broadreaching
chaos
network definition, in their excellent HTML document, What Is Chaos
Theory?, also available in a text only version.
Misconceptions
about chaos
theory
Chaos theory has received some attention,
beginning with
its popularity in movies such as Jurassic Park; public
awareness
of a science of chaos has been steadily increasing. However, as with
any
media covered item, many misconceptions have arisen concerning chaos
theory.
The most commonly held misconception about chaos
theory
is that chaos theory is about disorder. Nothing could be further from
the
truth! Chaos theory is not about disorder! It does not disprove
determinism or dictate that ordered systems are impossible; it does not
invalidate experimental evidence or claim that modelling complex
systems
is useless. The "chaos" in chaos theory is ordernot simply order,
but the very ESSENCE of order.
It is true that chaos theory dictates that minor
changes
can cause huge fluctuations. But one of the central concepts of chaos
theory
is that while it is impossible to exactly predict the state of a
system,
it is generally quite possible, even easy, to model the overall
behavior
of a system. Thus, chaos theory lays emphasis not on
the
disorder of the systemthe inherent unpredictability of a systembut
on the order inherent in the systemthe universal
behavior
of similar systems.
Thus, it is incorrect to say that chaos theory is
about
disorder. To take an example, consider Lorenz's Attractor. The Lorenz
Attractor
is based on three differential equations, three constants, and three
initial
conditions. The attractor represents the behavior of gas at any given
time,
and its condition at any given time depends upon its condition at a
previous
time. If the initial conditions are changed by even a tiny amount, say
as tiny as the inverse of Avogadro's number (a heinously small number
with
an order of 1E24), checking the attractor at a later time will yield
numbers
totally different. This is because small differences will propagate
themselves
recursively until numbers are entirely dissimilar to the original
system
with the original initial conditions.
However, the plot of the attractor will look very
much
the same.
Both systems will have totally different values
at any
given time, and yet the plot of the attractorthe overall behavior
of
the systemwill be the same.
Chaos theory predicts that complex nonlinear
systems
are inherently unpredictablebut, at the same time, chaos theory also
insures that often, the way to express such an unpredictable system
lies
not in exact equations, but in representations of the behavior of a
systemin
plots of strange attractors or in fractals. Thus, chaos theory,
which
many think is about unpredictability, is at the same time about
predictability
in even the most unstable systems.
How is chaos theory applicable to the real
world?
Everyone always wants to know one thing about new
discoverieswhat
good are they? So what good is chaos theory?
First and foremost, chaos theory is a theory. As
such,
much of it is of use more as scientific background than as direct
applicable
knowledge. Chaos theory is great as a way of looking at events which
happen
in the world differently from the more traditional strictly
deterministic
view which has dominated science from Newtonian times. Moviegoers who
watched
Jurassic Park are surely aware that chaos theory can profoundly affect
the way someone thinks about the world; and indeed, chaos theory is
useful
as a tool with which to interpret scientific data in new ways. Instead
of a traditional XY plot, scientists can now interpret phasespace
diagrams
whichrather than describing the exact position of some variable with
respect to timerepresents the overall behavior of a system. Instead
of
looking for strict equations conforming to statistical data, we can now
look for dynamic systems with behavior similar in nature to the
statistical
datasystems, that is, with similar attractors. Chaos theory provides
a sound framework with which to develop scientific knowledge.
However, this is not to say that chaos theory
has no applications
in real life.
Chaos theory techniques have been used to
model biological
systems, which are of course some of the most chaotic systems
imaginable.
Systems of dynamic equations have been used to modeleverything from population
growth to epidemics to arrhythmic
heart palpitations.
In fact, almost any chaotic system can be readily
modeledthe stock
market provides trends which can be analyzed with strange
attractors
more readily than with conventional explicit equations; a dripping
faucet
seems random to the untrained ear, but when plotted as a strange
attractor,
reveals an eerie order unexpected by conventional means.
Fractals have cropped up everywhere, most
notably
in graphic applications like the highly
successful
Fractal Design Painter series of products. Fractal image compression
techniques are still under research, but promise such amazing
results
as 600:1 graphic compression ratios. The movie special effects
industry
would have much less realistic clouds, rocks, and shadows without
fractal
graphic technology.
And of course, chaos theory gives people a
wonderfully
interesting way to become more interested in mathematics, one of the
more
unpopular pursuits of the day.
Who were the pioneers
of chaos
theory?
