Program in Applied and Computational Mathematics and
Department of Mechanical and Aerospace Engineering,
Through much of the eighteenth and nineteenth centuries, mathematics proceeded hand in hand with mechanics. The desire to solve problems terrestrial and celestial, including that of the stability of the solar system, drove advances in analysis, and new techniques were in turn applied to practical problems. Alas, the grand project to `integrate all the [differential] equations of mechanics' (L. Dirichlet) foundered on the rocks of nonlinearity, as Henri Poincar'e realised in 1888. However, Poincar'e also laid the foundations for the modern theory of dynamical systems. After a period of slow and fitful growth, mostly in ivory towers, this has variously ripened into nonlinear science and `chaos theory.'
I will sketch a history of dynamical systems from Poincar'e to Stephen Smale (the pre-color-graphics age), and through the rest of the century (the SIAM years), during which the abstract theory has been reunited with practice, and in doing so has influenced much of science and beyond. I will outline some of the major unifying ideas in the subject and the analytical techniques to which they have led, illustrating them by examples drawn from engineering and biology. In doing so, I will note some problems solved, partially solved, and some still strangely and attractively open.