  This paper considers continuously differentiable functions of two vector
variables that have (possibly a continuum of) min-max saddle points. We study
the asymptotic convergence properties of the associated saddle-point dynamics
(gradient-descent in the first variable and gradient-ascent in the second one).
We identify a suite of complementary conditions under which the set of saddle
points is asymptotically stable under the saddle-point dynamics. Our first set
of results is based on the convexity-concavity of the function defining the
saddle-point dynamics to establish the convergence guarantees. For functions
that do not enjoy this feature, our second set of results relies on properties
of the linearization of the dynamics, the function along the proximal normals
to the saddle set, and the linearity of the function in one variable. We also
provide global versions of the asymptotic convergence results. Various examples
illustrate our discussion.
