Historical Prologue
The theory of differential equations has a long history, beginning with Isaac Newton.
From the early Greeks through Copernicus, Kepler, and Galileo, the motions of
planets had been described directly in terms of their properties or characteristics,
for example, that they moved on approximately elliptical paths (or in combinations
of circular motions of different periods and amplitudes). Instead of this approach,
Newton described the laws that determine the motion in terms of the forces acting
on the planets. The effect of these forces can be expressed by differential equations.
The basic law he discovered was that the motion is determined by the gravitational
attraction between the bodies, which is proportional to the product of the two
masses of the bodies and one over the square of the distance between the bodies.
The motion of one planet around a sun obeying these laws can then be shown to lie
on an ellipse. The attraction of the other planets could then explain the deviation
of the motion of the planet from the elliptic orbit. This program was continued
by Euler, Lagrange, Laplace, Legendre, Poisson, Hamilton, Jacobi, Liouville, and
By the end of the nineteenth century, researchers realized that many nonlinear
equations did not have explicit solutions. Even the case of three masses moving
under the laws of Newtonian attraction could exhibit very complicated behavior
and an explicit solution was not possible (e.g., the motion of the sun, earth, and
moon cannot be given explicitly in terms of known functions). Short term solutions
could be given by power series, but these were not useful in determining long-term
behavior. Poincar´ e, working from 1880 to 1910, shifted the focus from finding
explicit solutions to discovering geometric properties of solutions. He introduced
many of the ideas in specific examples, which we now group together under the
heading of chaotic dynamical systems. In particular, he realized that a deterministic
system (in which the outside forces are not varying and are not random) can exhibit
behavior that is apparently random (i.e., it is chaotic).
In 1898, Hadamard produced a specific example of geodesics on a surface of
constant negative curvature which had this property of chaos. G. D. Birkhoff
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