78
3 Area-Preserving Maps
Fig. 3.18 The neighborhood
of the period 2 orbits of the
quadratic de Vogelaere map
just after they period-double
(p = −1.247) into four
period 4 orbits. (a) The
neighborhood of all four
period 4 points. (b)
Magnification of the
neighborhood of the good and
bad period 4 points
(daughters of the period 2
good point). (c)
Magnification of daughters of
the period 2 bad point
The actual value of p (k) and x (k) at the kth bifurcation for the first eight
bifurcations are shown in Table 3.1. In Fig. 3.19, we plot x (k) versus p (k) . Note
that the good point alternately lies to the left or right (on the symmetry line) of the
bifurcation point where it was born.
MacKay also found the convergence rate for 2 k -cycles perpendicular to the
symmetry line. By measuring the distance off the symmetry line of one of the