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 others. 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 xvii

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