Historical Origins of Chaos
Isaac Newton formulated his laws of motion as systems of differential
equations, with initial conditions. Mathematically, it can be shown
that solutions to such systems are in most cases unique.
Given a particular set of initial conditions,
the solution is determined uniquely.
(Check here
for a rigorous mathematical statement of uniqueness for initial
value problems.)
Another way of stating this is to say that
Given initial conditions are known (perfectly),
the solution is determined for all future tiimes (perfectly).
This reinforced the concept of "determinism" - that the future is an
inevitable consequence of the past.
The subtlety in this interpretation of a valid mathematical theory
is the assumption that initial conditions can ever be known "perfectly".
Another assumption is the implicit assumption of "continuous dependence"
- small changes in initial conditions lead to small changes in the solution.
The fact that large changes in the behavior of a complex system can depend on
infinitesimal changes in the initial state led to the discovery of
"chaotic systems".
One of the earliest physical systems in which chaos was "uncovered" was
the so called N-body problem. The actual application was the solar
system. The problem, as posed by King Oscar II of Sweden in 1887, was
to solve the equations governing the motion of N arbitrary planets.
In fact, the goal was to answer the question
Is our solar system stable for all time?
The French physicist and mathematician
Pierre Laplace (1749-1827) was one of the earliest scientists
who studied the stability of the solar
system. He showed that in fact it was indeed stable
for short times (ignoring the effects of tidal friction).
The French Mathematician
Henri Poincare considered a simple case (N=3) and showed that the
solution of 3 bodies could be extremely complex (non-periodic). He
submitted a memoir Sur le probleme des trois corps et les equations
de la dynamique to the Journal Acta Mathematica in 1890, and won the
prize. He showed that the solution of the 3-body problem was in general
unstable as time progressed.
The saving grace is that any instabilities in the solar system will
appear only after millions or even billions of years.
In 1927, Balthazar van der Pol discovered that the
equations governing vacuum tubes could exhibit "noisy" behavior.
Although it was not realized at the time, this was another example
of chaotic behavior.
Finally, in 1962,
Edward Lorenz discovered that a simple climate model demonstrated
extreme computational sensitivity. Restarting the computational model
with initial data rounded to several decimal places, resulted in
completely different behavior in time! This led to the first large
scale investigations into "deterministic chaos".