"If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon."Henri Poincaré in his 1903 essay "Science and Method"
A biography that gives more information about his other mathematical achievements besides chaos can be found in this nice History of Mathematics site under Poincaré.
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.
The idea that many simple nonlinear deterministic systems can behave in an apparently
unpredictable and chaotic manner was first noticed by the great French mathematician
Henri Poincaré.
In spite of this, the importance of chaos was not fully appreciated until the widespread
availability of digital computers
for numerical simulations and the demonstration of chaos in
various physical systems. This realization has broad implications for many fields of science,
and it is only within the past decade or so that the field
has undergone explosive growth. It is found that the ideas of chaos have been very fruitful
in such diverse disciplines as biology, economics, chemistry,
engineering, fluid mechanics, physics, just to name a few.
Chaos theory is among the youngest of the sciences, and has rocketed from its obscure
roots in the seventies to become one of the most fascinating fields in existence.
At the forefront of much research on physical systems, and already being implemented
in fields covering as diverse matter as arrhythmic pacemakers, image compression, and
fluid dynamics, chaos science promises to continue to yield absorbing scientific
information which may shape the face of science in the future.
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
order--not 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 system--the inherent unpredictability of a system--but on the
order inherent in the system--the universal behavior of similar
systems.
Chaos theory predicts that complex nonlinear systems are inherently
unpredictable--but, 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
system--in 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 discoveries--what 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 X-Y plot, scientists can now interpret phase-space diagrams
which--rather than describing the exact position of some variable with
respect to time--represents 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 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 model biological systems, which
are of course some of the most chaotic systems imaginable. Systems of
dynamic equations have been used to model everything from population growth
to epidemics to arrhythmic heart palpitations.
In fact, almost any chaotic system can be readily modeled--the 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.
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.