  Several algorithms are presented for the accurate computation of the leaves
in the foliation of an ODE near a hyperbolic fixed point. They are variations
of a contraction mapping method in [25] to compute inertial manifolds, which
represents a particular leaf in the unstable foliation. Such a mapping is
combined with one for the leaf in the stable foliation to compute the tracking
initial condition for a given solution. The algorithms are demonstrated on the
Kuramoto-Sivashinsky equation.
