Because we insist that all mesh points lie in fixed leaves, there is no accumulation of mesh points on one-dimensional submanifolds of . In this sense, our algorithm is independent of the dynamics on the circle. For the same reason it can be used to compute the two-dimensional unstable manifold of a hyperbolic fixed point .

Take a very small circle around in
and let *M* be its intersection with a
prescribed finite number of leaves of
, which we take to be equally spaced in
. The points in *M* now parametrize the
foliation, as it was the case in the previous section.
The linear approximation
is then immediate by defining as the
unit vector in pointing
away from ; see Figure 8.

With these definitions we can use the procedure GLOBALIZE to compute a prescribed number of rings of with prescribed tolerances , and . It is clear from the radial nature of the finite set of leaves that it is necessary to add leaves of during the computation to ensure that the maximal distance between neighboring mesh points does not become too big. (This effect is less pronounced for unstable manifolds of invariant circles of saddle-type; see Section 5.)

Written by: Bernd Krauskopf
& Hinke Osinga

Created: May 27 1997 ---
Last modified: Fri May 30 19:51:59 1997