3.3 The Standard Map
57
Fig. 3.7 In the neighborhood
of each homoclinic or
heteroclinic point, it is
possible to embed a Bernoulli
shift with an alphabet
containing an infinite number
of letters. Thus, the flow in
the neighborhood of the
homoclinic and heteroclinic
points is chaotic
homoclinic points about a given isolated hyperbolic fixed point. Let us select one
homoclinic point, r, and draw a small neighborhood, A r , at the point r so that one
side of A r lies along W (s) and another side lies along W (u) . Since r is a homoclinic
point, it and much of its neighborhood, A r , will be mapped to neighborhoods of the
hyperbolic fixed point, P , by T and T −1
. Let us assume that r and A r are mapped
to A o by T and are mapped to A 1 by T −1
. Because of the nature of the flow close
to the hyperbolic fixed point, strips of points U i that lie parallel to W (s) in A o will
be mapped to strips of points V j that lie parallel to W (u) in A 1 (see Fig. 3.7). The
net effect of this is that under repeated mappings T , strips U i that are parallel to
W (s) in A r get mapped back to A r by T but arrive as strips parallel to W (u) . Thus
a Bernoulli shift can be embedded in the flow in the neighborhood of each of the
homoclinic points. Furthermore, some of the sequences comprising this Bernoulli
shift will themselves be homoclinic points so the picture repeats itself infinitely
often in the neighborhood of each homoclinic point! In Sect. 2.7, we showed that
Bernoulli shifts have finite KS metric entropy and therefore are K-flows. Thus, the
flow in the neighborhood of each homoclinic and heteroclinic point is chaotic.
In subsequent sections, we only consider reversible twist maps. This includes
all maps that may be derived from a Hamiltonian that is even in momentum (Greene
1979a,b). A reversible twist map, T , is one that can be decomposed into a product
of involutions, S 1 and S 2 , such that T = S 2 S 1 , where S 2
1 = S 2
2 = I and I is the
identity map. This decomposition is important because the fixed points of I 1 and
I 2 form lines of symmetry of the map T , which greatly facilitate the study of fixed
points of the map.
3.3 The Standard Map
The standard map is a nonlinear twist map that is of special importance because it
describes the local behavior of nonintegrable dynamical systems in the separatrix
region of nonlinear resonances. From the standard map, we obtain a very clear
57
Fig. 3.7 In the neighborhood
of each homoclinic or
heteroclinic point, it is
possible to embed a Bernoulli
shift with an alphabet
containing an infinite number
of letters. Thus, the flow in
the neighborhood of the
homoclinic and heteroclinic
points is chaotic
homoclinic points about a given isolated hyperbolic fixed point. Let us select one
homoclinic point, r, and draw a small neighborhood, A r , at the point r so that one
side of A r lies along W (s) and another side lies along W (u) . Since r is a homoclinic
point, it and much of its neighborhood, A r , will be mapped to neighborhoods of the
hyperbolic fixed point, P , by T and T −1
. Let us assume that r and A r are mapped
to A o by T and are mapped to A 1 by T −1
. Because of the nature of the flow close
to the hyperbolic fixed point, strips of points U i that lie parallel to W (s) in A o will
be mapped to strips of points V j that lie parallel to W (u) in A 1 (see Fig. 3.7). The
net effect of this is that under repeated mappings T , strips U i that are parallel to
W (s) in A r get mapped back to A r by T but arrive as strips parallel to W (u) . Thus
a Bernoulli shift can be embedded in the flow in the neighborhood of each of the
homoclinic points. Furthermore, some of the sequences comprising this Bernoulli
shift will themselves be homoclinic points so the picture repeats itself infinitely
often in the neighborhood of each homoclinic point! In Sect. 2.7, we showed that
Bernoulli shifts have finite KS metric entropy and therefore are K-flows. Thus, the
flow in the neighborhood of each homoclinic and heteroclinic point is chaotic.
In subsequent sections, we only consider reversible twist maps. This includes
all maps that may be derived from a Hamiltonian that is even in momentum (Greene
1979a,b). A reversible twist map, T , is one that can be decomposed into a product
of involutions, S 1 and S 2 , such that T = S 2 S 1 , where S 2
1 = S 2
2 = I and I is the
identity map. This decomposition is important because the fixed points of I 1 and
I 2 form lines of symmetry of the map T , which greatly facilitate the study of fixed
points of the map.
3.3 The Standard Map
The standard map is a nonlinear twist map that is of special importance because it
describes the local behavior of nonintegrable dynamical systems in the separatrix
region of nonlinear resonances. From the standard map, we obtain a very clear