We consider a discrete dynamical system given by a diffeomorphism on . Such a map may be given explicitly, or be obtained as the Poincaré map of a vector field on a four-dimensional state space. Stable and unstable manifolds of invariant manifolds of saddle-type, here fixed points or invariant circles, play important roles in organizing the global dynamics. It is well-known that the stable manifolds form boundaries between different basins of attraction. Furthermore, the transverse intersection of stable and unstable manifolds leads to homoclinic or heteroclinic tangle, associated with chaos. These manifolds are global, often noncompact objects that can have very complicated structure. Only in special situations it is possible to find stable and unstable manifolds analytically. In general they need to be computed with numerical methods.

In this paper we compute the two-dimensional
unstable manifold
of an invariant manifold of saddle-type
of a three-dimensional diffeomorphism.
(The stable manifold
can be computed by considering the inverse.)
There are two different cases: the unstable manifold
of a hyperbolic fixed point
, and the unstable manifold of a normally hyperbolic
invariant circle *H*.
The Unstable Manifold Theorem
[Hirsch et al. 1977,
Palis and De Melo 1982]
guarantees the existence of
the * local unstable manifold*
in a neighborhood of the invariant
manifold of saddle-type. The two cases are sketched in
Figure 1. See also the
animations.

The idea is to obtain the global unstable manifold
by * globalizing* the
local unstable manifold .
In practice, we
reliably compute a sufficient piece of ,
so that conclusions on the global dynamics can
be drawn.
The unstable manifold
is represented by a discrete set of mesh points.
The quality of the mesh can be
prescribed, and the mesh is adapted whenever necessary.
Our algorithm was originally
developed for the computation of unstable manifolds
of a normally hyperbolic invariant
circle of saddle-type. With a slight adaptation,
it can also be used to find a two-dimensional unstable
manifold of a hyperbolic
fixed point.

As starting data for our algorithm we need an approximation
of the local unstable manifold .
There are different algorithms
for computing of a hyperbolic fixed
point ;
see [Osinga 1996] and biblio therein.
The work presented here was motivated
by the possibility of
obtaining the starting data for the
computation of the unstable manifold of an
invariant circle *H* of saddle-type
in the form of a
linear approximation of
with the method in
[Osinga 1996,
Broer et al. 1996,
Broer et al. 1997]. Their method is a variation of the graph
transform that allows the computation
of a normally hyperbolic
invariant circle of saddle-type of a three-dimensional map.
The key idea is to start with a known invariant circle *H* of
a map *f* together with the *Df*-invariant splitting of the tangent
space at *H*. This splitting induces *Df*-invariant stable and
unstable normal bundles that are embedded in in a neighborhood
of *H*. Because *H* is normally hyperbolic, these embedded normal
bundels form a well-defined coordinate system in a neighborhood of
*H*. The invariant circle
of a small perturbation of *f* is
computed as the graph over the known circle *H* in the coordinate
system given by the embedded normal bundles. As a special feature of
the method
the new -invariant splitting of the invariant circle
is computed in a second step,
regardless of the dynamics on
. Consequently, the method can be used in
a continuation setting: a known invariant circle can be
followed by increasing in small steps. The embedded unstable
normal bundle of
[Osinga 1996,
Broer et al. 1996,
Broer et al. 1997]
is the first order approximation of , the local
unstable manifold of *H*.

This allows us to globalize this first order approximation to
compute a significant piece of .
To be more concrete, the invariant
circle *H* is known in a
finite mesh *M* of points, and at each mesh
point we are
given the normal direction of the embedded normal bundle.
(Since is of saddle-type,
is a vector.)
We now choose a linear foliation
of the state space so that each
leaf has a
unique intersection with *H*.
From the unstable normal bundle above
we compute
unit vectors
for all , such that is tangent to . Starting from the linear
approximation given by *M* and ,
we compute the
intersection of the unstable manifold with the
finitely many leaves
of .
For this to work we need the following.

** Foliation Condition**

In each leaf of there is a
unique curve of intersection with the unstable manifold
.
In other words, the unstable manifold intersects each leaf
transversally.

Assuming this foliation condition is satisfied, we can compute the unstable manifold in each of the leaves of as a sequence of points that have a prescribed distance from each other. The set of sequences in a finite number of leaves defines a mesh that represents the unstable manifold. This computation can be done in steps by adding rings or bands, that is, by adding a single new point to the sequence in each leaf. In this way, one can see the unstable manifold grow during the computation.

The linear foliation has no dynamical property, but should be seen as an a priori definition of the mesh. By adding additional leaves during the computation we guarantee the quality of the mesh on the unstable manifold. By construction, our method is independent of the dynamics on the invariant circle. We can use it for the computation of the two-dimensional unstable manifold of a fixed point, if we interpret the fixed point as an invariant circle in polar coordinates. A detailed description of the algorithm can be found in Section 3.

The procedure of adding rings or bands fails when the computed portion of the unstable manifold no longer intersects each leaf of in a unique curve. A piece of the intersection is then missed in the computation, and the algorithm is lacking information about a part of the unstable manifold. Because this information is necessary in the computation, the algorithm stops. This is discussed in more detail in Section 6; see also Figure 6 and Figure 18. Note that by definition the foliation is transverse to so that at least a part of can be computed. Furthermore, we think that many interesting examples satisfy the foliation condition; see Section 5. Possible relaxations of the foliation condition are discussed in Section 6.

In summary, we present an algorithm that computes a growing piece of the two-dimensional unstable manifold of a normally hyperbolic invariant circle of saddle-type or a hyperbolic saddle point, until the unstable manifold becomes tangent to the foliation . The mesh representing the invariant manifold is of a prescribed quality. This allows us to study the global dynamics of many interesting systems, in particular those that satisfy the Foliation Condition.

This paper is organized as follows. In Section 2 we give an overview over the literature on the calculation of global unstable manifolds. In Section 3 we give a detailed description of the algorithm, and Section 4 deals with its correctness. The performance of our method is demonstrated in Section 5 with a number of examples. The applicability of the algorithm and some open problems are discussed in Section 6.

Written by: Bernd Krauskopf
& Hinke Osinga

Created: May 27 1997 ---
Last modified: Wed Jul 2 10:50:50 1997