of a map f
by
and
These manifolds are tangent to the eigenvectors of f at .
We are led to study a two-dimensional map f by considering the Poincaré map of a three-dimensional flow in the vicinity of a periodic orbit.
Figure: (a) Poincaré map in the vicinity of a periodic orbit,
. (b) The map shown with a transversal intersection at a homoclinic
point.
This situation is illustrated in Figure 4.13. The map f is
orientation
preserving
because it comes from a smooth flow.
The periodic orbit of the flow gives rise to the fixed point of the
map.
The fixed point has a one-dimensional stable manifold 
and a one-dimensional unstable manifold . Poincaré was led
to his discovery of chaotic behavior and homoclinic tangles
(see section 3.6 and Appendix H) by considering the interaction between
the stable and unstable manifold of .
One possible interaction is shown in Figure 4.13(a) where
the unstable manifold exactly matches the stable manifold,
.
However, such a smooth match is exceptional. The more common possibility is for a transversal intersection between the stable and unstable manifold (Fig. 4.13(b)). The location of the transversal intersection is called a homoclinic point when both the unstable and stable manifold emanate from the same periodic orbit (Fig. 4.14(a)).
Figure 4.14: (a) A homoclinic point. (b) A heteroclinic point.
The intersection point is called a
heteroclinic point when the manifolds emanate from
different periodic orbits.
A heteroclinic point is shown in Figure 4.14(b)
where the unstable manifold emanating from intersects the
stable manifold of a different fixed point 
.
The existence of a single homoclinic or heteroclinic point forces the existence of an infinity of such points. Moreover, it also gives rise to a homoclinic (heteroclinic) tangle . This tangle is the geometric source of chaotic motions.
Figure 4.15: The interaction of the stable manifold and the
unstable manifold with a homoclinic point 
.
The homoclinic point gets mapped to the homoclinic point 
.
The orientation of the map is determined by considering where points a and b
in the vicinity of are mapped.
To see why this is so, consider Figure 4.15. A homoclinic point
is indicated at . This homoclinic point is part of both
the stable manifold and the unstable manifold,
Also shown is a point a that lies on the stable manifold
behind (the direction is determined by the arrow on the
manifold), i.e., on 
. Similarly, the point
b lies on the unstable manifold with 
on 
.
Now, we must try to find the location of the next iterate
of subject to the following conditions:
(all the iterates of a homoclinic point are also homoclinic
points).
. must lie at a new homoclinic point (that is, at a new
intersection point) ahead of . The first candidate for the location
of is the next intersection point, indicated at d. However,
could not be located here because that would imply that the map f is
orientation reversing (see Prob. 4.21). The next possible location, which does
satisfy all the above conditions, is indicated by . More complicated
constructions could be envisioned that are consistent with the above
conditions, but the solution shown in Figure 4.15 is the simplest.
Now is itself a homoclinic point.
And the same argument applies again: the point 
must lie closer
to and
ahead of (Fig. 4.16(a)).
Figure 4.16: (a) Images and preimages of a homoclinic point. (b)
A homoclinic tangle resulting from a single homoclinic point.
In this way a single homoclinic orbit must generate an infinite number
of homoclinic orbits. This sequence of homoclinic points asymptotically
approaches .
Since f arises from a flow, it is a diffeomorphism and
thus invertible. Therefore, exactly the same argument applies
to the preimages of 
. That is, approaches
via the unstable manifold. The end result of this construction
is the
violent oscillation of and in the region of .
These oscillations form the homoclinic tangle indicated schematically
in Figure 4.16(b).
The situation is even more complicated than it initially appears. The homoclinic points are not periodic orbits, but Birkhoff and Smith showed that each homoclinic point is an accumulation point for an infinite family of periodic orbits [9]. Thus, each homoclinic tangle has an infinite number of homoclinic points, and in the vicinity of each homoclinic point there exists an infinite number of periodic points. Clearly, one major goal of dynamical systems theory, and nonlinear dynamics, is the development of techniques to dissect and classify these homoclinic tangles. In section 4.8 we will show how the orbit structure of a homoclinic tangle is organized by using a horseshoe map. In Chapter 5 we will continue this topological approach by showing how knot theory can be used to unravel a homoclinic tangle.