Douglas S. Shafer
Professor of Mathematics
University of North Carolina at Charlotte
The title of a section of a popular introductory textbook on differential
equations contains the phrase ``why we cannot solve very many differential
equations." In the face of this fact, one alternative is to resort to
numerical methods of approximating solutions. Another, begun in earnest in
the latter part of the nineteenth century and a subject of active current
research, is to attach to a differential equation, or to a family of
differential equations depending on parameters, a geometric object termed
the ``phase portrait," and to gain insight into solutions of the family, their
structure, long-term behavior, and metamorphosis under alteration of the
parameters, by studying the phase portrait. In this talk I will introduce
these ideas in the context of such familiar settings as the pendulum, and
give a glimpse into the mixture of techniques, both analytic and geometric,
that are used, and the kind of results that are obtained.