What is chaos theory?
Formally, chaos theory is defined as the study of
complex nonlinear dynamic systems. Complex implies just
implies recursion and higher mathematical algorithms, and dynamic
implies nonconstant and nonperiodic. Thus chaos theory is, very
the study of forever changing complex systems based on mathematical
of recursion, whether in the form of a recursive process or a set of
equations modeling a physical system.
For a more rigorous definition of chaos theory,
advisable to visit the much more scientific, much more broad-reaching
network definition, in their excellent HTML document, What Is Chaos
Theory?, also available in a text only version.
Chaos theory has received some attention,
its popularity in movies such as Jurassic Park; public
of a science of chaos has been steadily increasing. However, as with
media covered item, many misconceptions have arisen concerning chaos
The most commonly held misconception about chaos
is that chaos theory is about disorder. Nothing could be further from
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
is useless. The "chaos" in chaos theory is order--not simply order,
but the very ESSENCE of order.
It is true that chaos theory dictates that minor
can cause huge fluctuations. But one of the central concepts of chaos
is that while it is impossible to exactly predict the state of a
it is generally quite possible, even easy, to model the overall
of a system. Thus, chaos theory lays emphasis not on
disorder of the system--the inherent unpredictability of a system--but
on the order inherent in the system--the universal
of similar systems.
Thus, it is incorrect to say that chaos theory is
disorder. To take an example, consider Lorenz's Attractor. The Lorenz
is based on three differential equations, three constants, and three
conditions. The attractor represents the behavior of gas at any given
and its condition at any given time depends upon its condition at a
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
an order of 1E-24), checking the attractor at a later time will yield
totally different. This is because small differences will propagate
recursively until numbers are entirely dissimilar to the original
with the original initial conditions.
However, the plot of the attractor will look very
Both systems will have totally different values
given time, and yet the plot of the attractor--the overall behavior
the system--will be the same.
Chaos theory predicts that complex nonlinear
are inherently unpredictable--but, at the same time, chaos theory also
insures that often, the way to express such an unpredictable system
not in exact equations, but in representations of the behavior of a
plots of strange attractors or in fractals. Thus, chaos theory,
many think is about unpredictability, is at the same time about
in even the most unstable systems.
How is chaos theory applicable to the real
Everyone always wants to know one thing about new
good are they? So what good is chaos theory?
First and foremost, chaos theory is a theory. As
much of it is of use more as scientific background than as direct
knowledge. Chaos theory is great as a way of looking at events which
in the world differently from the more traditional strictly
view which has dominated science from Newtonian times. Moviegoers who
Jurassic Park are surely aware that chaos theory can profoundly affect
the way someone thinks about the world; and indeed, chaos theory is
as a tool with which to interpret scientific data in new ways. Instead
of a traditional X-Y plot, scientists can now interpret phase-space
which--rather than describing the exact position of some variable with
respect to time--represents the overall behavior of a system. Instead
looking for strict equations conforming to statistical data, we can now
look for dynamic systems with behavior similar in nature to the
data--systems, 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
systems, which are of course some of the most chaotic systems
Systems of dynamic equations have been used to modeleverything from population
growth to epidemics to arrhythmic
In fact, almost any chaotic system can be readily
market provides trends which can be analyzed with strange
more readily than with conventional explicit equations; a dripping
seems random to the untrained ear, but when plotted as a strange
reveals an eerie order unexpected by conventional means.
Fractals have cropped up everywhere, most
in graphic applications like the highly
Fractal Design Painter series of products. Fractal image compression
techniques are still under research, but promise such amazing
as 600:1 graphic compression ratios. The movie special effects
would have much less realistic clouds, rocks, and shadows without
And of course, chaos theory gives people a
interesting way to become more interested in mathematics, one of the
unpopular pursuits of the day.
Who were the pioneers