3.3 The Standard Map
61
Fig. 3.10 Some orbits of the
standard map (with periodic
boundary conditions):
(a) K = 0.1716354; (b)
K = 0.4716354 (Plots by
Steve Cocke)
Figure 3.11d shows the standard map at the critical value, K ∗ = 0.9716354,
of the parameter K. At this value of K, the last remaining KAM torus, which
stretches horizontally from x = 0 to x = 1.0, is broken. This torus is easily seen
in Fig. 3.11d. For values of K ≤ 0.9716354, it is not possible for any trajectories
to diffuse vertically through this line and reach arbitrarily large values of p. We
see that all of the separatrices shown appear to be chaotic. In Figs. 3.12e and f, we
show the structure of the standard map for K = 1.1716354 and K = 3.9716254,
respectively. For K = 1.1716354, the primary resonances are still intact, although
they are surrounded by a chaotic sea. For K = 3.9716354, almost all structure is
wiped out.
61
Fig. 3.10 Some orbits of the
standard map (with periodic
boundary conditions):
(a) K = 0.1716354; (b)
K = 0.4716354 (Plots by
Steve Cocke)
Figure 3.11d shows the standard map at the critical value, K ∗ = 0.9716354,
of the parameter K. At this value of K, the last remaining KAM torus, which
stretches horizontally from x = 0 to x = 1.0, is broken. This torus is easily seen
in Fig. 3.11d. For values of K ≤ 0.9716354, it is not possible for any trajectories
to diffuse vertically through this line and reach arbitrarily large values of p. We
see that all of the separatrices shown appear to be chaotic. In Figs. 3.12e and f, we
show the structure of the standard map for K = 1.1716354 and K = 3.9716254,
respectively. For K = 1.1716354, the primary resonances are still intact, although
they are surrounded by a chaotic sea. For K = 3.9716354, almost all structure is
wiped out.