### Invited Presentation: How Nonlinear is Science? Reflections on a Chaotic, Dynamical Century

**Philip Holmes**

*Program in Applied and Computational Mathematics and
*

Department of Mechanical and Aerospace Engineering,

Princeton University

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.