3.2 Twist Maps
55
Fig. 3.4 The flow of points in the neighborhood of fixed points. For regular hyperbolic points
(residue R < 0), successive points on an orbit remain on one side of the fixed point, while for
inversion hyperbolic points (residue R > 1) successive points alternate across the fixed point. The
numbers indicate the sequence in time of the points (the residue, R, is defined in Sect. 3.3)
For −2 < t < 2, the eigenvalues form complex conjugate pairs that lie on the
unit circle, and the fixed points are elliptic. For t > 2, the fixed point is regular
hyperbolic. For t < −2, the fixed point is inversion hyperbolic (subsequent points
of the mapping alternate across the fixed point (see Fig. 3.4). For the special cases
t = ±2, the eigenvalues are degenerate, having values +1 or −1, and the fixed point
is parabolic. Parabolic fixed points are generally unstable (MacKay 1982).
If the mapping is defined in terms of smooth continuous functions, the eigenvectors of ∇T M
in the neighborhood of the fixed point will be smooth and continuous.
For elliptic fixed points, the eigenvalues will be pure imaginary and the eigenvectors
will describe motion that oscillates about the fixed point. For hyperbolic fixed points,
the eigenvalues will be real and of the form λ 1 =
1
λ and λ 2 = λ, where λ is real and
λ > 1. Let us denote the eigencurve associated with the eigenvalue
1
λ as W (s) and
the eigencurve associated with the eigenvalue λ as W (u) .
Once the eigencurves of the tangent map have been found, they can be extended
away from the neighborhood of the fixed point by using the full map, T M
. These
extensions of the eigencurves are also denoted W (s) and W (u) , and are called stable
manifolds and unstable manifolds, respectively. Points on the stable manifolds,
W (s) , will be mapped toward the fixed point since (∇T M
) n W (s) = (
1
λ ) n W (s) , while
points on the unstable manifolds, W (u) , will be mapped away from the fixed point
since (∇T M
) n W (u) = λ n W (u) (see Fig. 3.5).
3.2.5 Homoclinic and Heteroclinic Points
For integrable systems, the stable and unstable manifolds of one fixed point will
join smoothly together (Fig. 3.5a) or will connect smoothly to those of another fixed
point (Fig. 3.5b). For example, the point r in Fig. 3.5b will be mapped toward the
fixed point P by T o but toward Q by T −1
o .
55
Fig. 3.4 The flow of points in the neighborhood of fixed points. For regular hyperbolic points
(residue R < 0), successive points on an orbit remain on one side of the fixed point, while for
inversion hyperbolic points (residue R > 1) successive points alternate across the fixed point. The
numbers indicate the sequence in time of the points (the residue, R, is defined in Sect. 3.3)
For −2 < t < 2, the eigenvalues form complex conjugate pairs that lie on the
unit circle, and the fixed points are elliptic. For t > 2, the fixed point is regular
hyperbolic. For t < −2, the fixed point is inversion hyperbolic (subsequent points
of the mapping alternate across the fixed point (see Fig. 3.4). For the special cases
t = ±2, the eigenvalues are degenerate, having values +1 or −1, and the fixed point
is parabolic. Parabolic fixed points are generally unstable (MacKay 1982).
If the mapping is defined in terms of smooth continuous functions, the eigenvectors of ∇T M
in the neighborhood of the fixed point will be smooth and continuous.
For elliptic fixed points, the eigenvalues will be pure imaginary and the eigenvectors
will describe motion that oscillates about the fixed point. For hyperbolic fixed points,
the eigenvalues will be real and of the form λ 1 =
1
λ and λ 2 = λ, where λ is real and
λ > 1. Let us denote the eigencurve associated with the eigenvalue
1
λ as W (s) and
the eigencurve associated with the eigenvalue λ as W (u) .
Once the eigencurves of the tangent map have been found, they can be extended
away from the neighborhood of the fixed point by using the full map, T M
. These
extensions of the eigencurves are also denoted W (s) and W (u) , and are called stable
manifolds and unstable manifolds, respectively. Points on the stable manifolds,
W (s) , will be mapped toward the fixed point since (∇T M
) n W (s) = (
1
λ ) n W (s) , while
points on the unstable manifolds, W (u) , will be mapped away from the fixed point
since (∇T M
) n W (u) = λ n W (u) (see Fig. 3.5).
3.2.5 Homoclinic and Heteroclinic Points
For integrable systems, the stable and unstable manifolds of one fixed point will
join smoothly together (Fig. 3.5a) or will connect smoothly to those of another fixed
point (Fig. 3.5b). For example, the point r in Fig. 3.5b will be mapped toward the
fixed point P by T o but toward Q by T −1
o .