To construct a cross section for nonplanar periodic motion we could imagine
a plot of the position of the center of the string, and
the forcing phase, 
(Fig.\
3.18). A map can be associated to an orbit in 
by
recording the position of the orbit once each forcing period.
Although this map is not a true Poincaré map, it is easy to obtain
experimentally and will be useful in explaining the notion of a torus
attractor (see section 3.8.1).
An elliptical periodic orbit in the flow is represented in this cross section
by a discrete set of points that lie on a closed curve. This curve is
topologically a circle, 
(Fig. 3.18(b)).
Figure 3.18: Experimental cross section for the nonplanar string vibrations.
Similarly, a precessing ellipse (quasiperiodic motion) can generate an infinite number of points; these points fill out this circle (Fig. 3.18(c)).
In the extended space 
, this quasiperiodic motion represents
a dense winding of a torus as shown in Figure 3.18(d). Topologically, a
torus is a space
constructed from the Cartesian product of two circles,
. In general, an n torus is constructed from
n copies of a circle,
and a torus attractor naturally arises whenever
quasiperiodic motion is encountered in a dissipative dynamical
system. The
torus is an attractor because it is an invariant set and an attracting limit
set. This is illustrated in Figure 3.19, which shows how orbits
are attracted to a torus.
A graph of a quasiperiodic orbit on a torus attractor is an amplitude-modulated time series (Fig. 3.12).