56
3 Area-Preserving Maps
Fig. 3.5 The stable and unstable manifolds, W s and W u , of hyperbolic fixed points for integrable
systems join smoothly. (a) A point r on W (s) and W (u) is mapped to the same fixed point, P , by
T o and T −1
o . (b) A point r on W (s) and W (u) is mapped to the fixed point P by T o and to the fixed
point Q by T −1
o
Fig. 3.6 For nonintegrable systems, the stable and unstable manifolds no longer join smoothly but
oscillate and intersect transversally. (a) For stable and unstable manifolds that approach the same
fixed point, the intersections, r, r , r , etc., are called homoclinic points. (b) For manifolds that
approach different hyperbolic fixed points, the points of transversal intersection, r, r , r , etc., are
called heteroclinic points
For a nonintegrable system, a totally different behavior occurs. As one begins
to perturb the map, the stable and unstable manifolds, W (s) and W (u) , respectively,
begin to oscillate and intersect one another transversally at an infinite number of
places (see Fig. 3.6). For the case in Fig. 3.6a, where W (s) and W (u) belong to the
same hyperbolic fixed point, P , the points of intersection, r, r , r , etc., are called
homoclinic points, while for the case in Fig. 3.6b, where W (s) and W (u) attach to
separate hyperbolic fixed points, P and Q, the points of intersection, r, r , r , etc.,
are called heteroclinic points. Homoclinic points in Fig. 3.5a are mapped toward
P by T and T −1
, but in opposite directions, while heteroclinic points in Fig. 3.6b
are mapped toward Q by T and toward P by T −1
. The homoclinic or heteroclinic
points become more and more closely spaced as one approaches the hyperbolic fixed
points and therefore, since area must be preserved by the map, the oscillations must
grow in amplitude as one approaches the hyperbolic fixed points. It has been shown
that it is possible to embed a Bernoulli shift with an alphabet containing an infinite
number of “letters” (the baker’s map has two letters) in the neighborhood of each
homoclinic or heteroclinic point.
A very good discussion of the embedding of the Bernoulli shift into the
neighborhood of the homoclinic and heteroclinic points has been given by Moser
(1973). Moser draws a picture similar to Fig. 3.7, which shows a few of the
3 Area-Preserving Maps
Fig. 3.5 The stable and unstable manifolds, W s and W u , of hyperbolic fixed points for integrable
systems join smoothly. (a) A point r on W (s) and W (u) is mapped to the same fixed point, P , by
T o and T −1
o . (b) A point r on W (s) and W (u) is mapped to the fixed point P by T o and to the fixed
point Q by T −1
o
Fig. 3.6 For nonintegrable systems, the stable and unstable manifolds no longer join smoothly but
oscillate and intersect transversally. (a) For stable and unstable manifolds that approach the same
fixed point, the intersections, r, r , r , etc., are called homoclinic points. (b) For manifolds that
approach different hyperbolic fixed points, the points of transversal intersection, r, r , r , etc., are
called heteroclinic points
For a nonintegrable system, a totally different behavior occurs. As one begins
to perturb the map, the stable and unstable manifolds, W (s) and W (u) , respectively,
begin to oscillate and intersect one another transversally at an infinite number of
places (see Fig. 3.6). For the case in Fig. 3.6a, where W (s) and W (u) belong to the
same hyperbolic fixed point, P , the points of intersection, r, r , r , etc., are called
homoclinic points, while for the case in Fig. 3.6b, where W (s) and W (u) attach to
separate hyperbolic fixed points, P and Q, the points of intersection, r, r , r , etc.,
are called heteroclinic points. Homoclinic points in Fig. 3.5a are mapped toward
P by T and T −1
, but in opposite directions, while heteroclinic points in Fig. 3.6b
are mapped toward Q by T and toward P by T −1
. The homoclinic or heteroclinic
points become more and more closely spaced as one approaches the hyperbolic fixed
points and therefore, since area must be preserved by the map, the oscillations must
grow in amplitude as one approaches the hyperbolic fixed points. It has been shown
that it is possible to embed a Bernoulli shift with an alphabet containing an infinite
number of “letters” (the baker’s map has two letters) in the neighborhood of each
homoclinic or heteroclinic point.
A very good discussion of the embedding of the Bernoulli shift into the
neighborhood of the homoclinic and heteroclinic points has been given by Moser
(1973). Moser draws a picture similar to Fig. 3.7, which shows a few of the