  We develop a validated numerical procedure for continuation of local
stable/unstable manifold patches attached to equilibrium solutions of ordinary
differential equations. The procedure has two steps. First we compute an
accurate high order Taylor expansion of the local invariant manifold. This
expansion is valid in some neighborhood of the equilibrium. An important
component of our method is that we obtain mathematically rigorous lower bounds
on the size of this neighborhood, as well as validated a-posteriori error
bounds for the polynomial approximation. In the second step we use a rigorous
numerical integrating scheme to propagate the boundary of the local
stable/unstable manifold as long as possible, i.e.\ as long as the integrator
yields validated error bounds below some desired tolerance. The procedure
exploits adaptive remeshing strategies which track the growth/decay of the
Taylor coefficients of the advected curve. In order to highlight the utility of
the procedure we study the embedding of some two dimensional manifolds in the
Lorenz system.
