We explain the algorithm for computing a two-dimensional unstable manifold of a normally hyperbolic invariant circle of saddle-type. How this algorithm can be used for the case of a hyperbolic fixed point is shown in Section 3.3. We begin by introducing some notation.

Let be a diffeomorphism
with a normally hyperbolic invariant circle *H* of saddle-type, and
let be the unstable manifold of *H*. We assume that *f* can
be transformed to a function defined on , where
is identified with , such that *H* is parametrized over
. Then there exist a global linear foliation that
foliates .

The invariant circle *H* can be computed by the method in
[Osinga 1996,
Broer et al. 1996,
Broer et al. 1997]. It
is represented by a finite
mesh *M* of points on *H*.
The number of points in *M* can be prescribed.
Suppose that the linear foliation satisfies the Foliation
Condition. (This is always true in a neighborhood of *H*; see also
Section 6.) Recall that this means
that intersects each leaf
in a unique curve.
Our goal is to compute an approximation of the unique
intersection curve
for the finitely many leaves of
.

To start the algortihm
we need to know the
first order approximation of
in each leaf of .
It is defined by
unit vectors that are tangent to .
The vectors
can be obtained
by interpolation
from the embedded unstable normal bundle that can be obtained
from the *Df*-invariant splitting that is
used in the method of
[Osinga 1996,
Broer et al. 1996,
Broer et al. 1997]; see
Figure 3.
From now on we assume that the starting
data for the algorithm, namely *M*,
and , are known.

3.1 Iterating a fundamental domain

3.2 Globalization by adding discrete circles

3.3 The case of a hyperbolic fixed point

3.4 Mesh adaptation

Written by: Bernd Krauskopf
& Hinke Osinga

Created: May 27 1997 ---
Last modified: Fri May 30 19:49:09 1997