  We present several topics involving the computation of dynamical systems. The
emphasis is on work in progress and the presentation is informal -- there are
many technical details which are not fully discussed. The topics are chosen to
demonstrate the various interactions between numerical computation and
mathematical theory in the area of dynamical systems. We present an algorithm
for the computation of stable manifolds of equilibrium points, describe the
computation of Hopf bifurcations for equilibria in parametrized families of
vector fields, survey the results of studies of codimension two global
bifurcations, discuss a numerical analysis of the Hodgkin and Huxley equations,
and describe some of the effects of symmetry on local bifurcation.
