  We give a surface for which the Ricci Flow applied to the metric will
increase the topological entropy of the geodesic flow. Specifically, we first
adapt the Melnikov method to apply to a Ricci Flow perturbation and then we
construct a surface which is closely related to a surface of revolution, but
does not quite have rotational symmetry. This is done by adapting the Liouville
metric representation of a surface of revolution. The final steps of the
Melnikov method require numerical integration.
