Title:  Codimension-one persistence beyond all orders of
        reversible homoclinic orbits to degenerate saddle centres.

Speaker: Alan Champneys, Bristol, UK

This talk follows on from part of the lecture given by Lombardi.
Specifically we will consider numerically the question of persistence
of homoclinic solutions to the origin of a reversible system undergoing
a singluar saddle centre local bifurcation. By this we mean an
equilibrium in four-dimensions whose linearisation has eigenvlaues
+lambda, -lambda, +i omega, -i omega in the limit as lambda tends
to zero. Lombardi's analysis shows that generically a small
homoclinic solution in the `slow' hyperbolic part of the vector field
does not survive. Numerical evidence points to a certain codimension-one 
persistence result treating lambda and omega as independent parameters
(with nonlinearity fixed). Specifically, the numerics support the conjecture
that given a sign condition on a term in the normal
form, homoclinic orbits `bifurcate' from the singular limit lambda=0
at a regular sequence of omega values. A comprehensive study is carried
out of a class of reversible system that can be written as a fourth-order
equation. This has application to solitary water waves with surface
tension and to recently discovered `embedded solitons' in nonlinear optics.