The standard approach for globalizing one-dimensional unstable
manifolds is to start with a point on the linear approximation,
take the line segment between this point and its *f*-image, and
compute iterates of this
(approximation of a) fundamental domain. For the two-dimensional
version of this method, one would take a circle of points
on the linear
approximation. If the circle is close enough to *H*,
one may assume that the
*f*-image of this circle is almost a circle as well. Therefore,
one obtains an annulus as a
fundamental domain.

Since the fundamental domain cannot be iterated as a
continuous object, one needs to discretize it.
As was pointed out earlier, merely iterating the
points of the discretiation will lead to a possible accumulation of
points on a one-dimensional submanifold of
; see also
Figure 2.
In our situation, we can avoid this accumulation by keeping track
of where the iterates of the circle intersect
.
However, there is a second problem. Away from
*H*, the iterates of the initial circle on the linear approximation of
may not look like circles
at all. In practice, already after
a few iterations, the successive images of the fundamental domain
may have a very `unpractical' shape.
For example in the *3D*-fattened Arnold
family in Section 5.1, the invariant circle contains two
fixed points. Close to the one-dimensional stable manifold of the fixed
point that is repelling on *H*, the *f*-image of an annulus grows a
`nose' that stretches and winds around the circle
several times, but at least once. This is
unpractical, because it is unclear how one can obtain a nice mesh
on the unstable manifold in this situation. It may look like the
unstable manifold has holes.

In order to avoid such problems we drop the idea of iterating a fundamental domain. Instead we let the manifold grow at the same speed in each direction, which means that we add rings of fixed width . How this can be done is explained in the following section.

Written by: Bernd Krauskopf
& Hinke Osinga

Created: May 27 1997 ---
Last modified: Fri May 30 19:49:51 1997