A unified approach to compute foliations, inertial manifolds, and   tracking initial conditions
  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. 
Y.-M. Chung and M. S. Jolly
math.NA math.DS