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For the sake of definiteness, let us fix the normalized amplitude and frequency of the external
drive to be and , respectively.^{} Furthermore, let us investigate any changes which
may develop in the nature of the
pendulum's timeasymptotic motion
as the qualityfactor is varied. Of course, if is made sufficiently small
(i.e., if the pendulum is embedded in a sufficiently viscous medium) then we expect the
amplitude of the pendulum's timeasymptotic motion to become low enough that the linear analysis
outlined in Section 15.3 is valid. Indeed, we expect nonlinear effects to manifest themselves
as is gradually made larger, and the amplitude of the pendulum's motion
consequently increases to such an extent that the
small angle approximation breaks down.
Figure 62:
Equally spaced (in time) points on a timeasymptotic orbit
in phasespace. Data calculated numerically
for , , , , and
.

Figure 62 shows a timeasymptotic orbit in phasespace
calculated numerically for a case where is sufficiently
small (i.e., ) that the small angle approximation holds reasonably well. Not surprisingly,
the orbit is very similar to the analytic orbits
described in Section 15.3. The fact that the orbit consists
of a single loop, and forms a closed curve in phasespace,
strongly suggests that the corresponding
motion is periodic with the same period as the external drivewe term this type of motion
period1 motion. More generally, period motion consists of motion which
repeats itself exactly every periods of the external drive (and, obviously,
does not repeat itself on any timescale less than periods). Of course, period1 motion
is the only allowed timeasymptotic motion in the small angle limit.
It would certainly be helpful
to possess a graphical test for period motion. In fact, such a test was developed more than a
hundred years ago by the
French mathematician Henry Poincaré. Nowadays, it is called a
Poincaré section in his honour. The idea of a Poincaré section, as applied
to a periodically driven pendulum, is very simple.
As before, we calculate the timeasymptotic motion of the pendulum, and visualize it as a
series of points in 
phasespace. However, we only plot one point per period of the external drive. To be more
exact, we only plot a point when

(1250) 
where is any integer, and is referred to as the Poincaré phase.
For period1 motion, in which the motion repeats itself exactly every period of the
external drive, we expect the Poincaré section to consist of only one point
in phasespace (i.e., we expect all of the points to plot on top of one another).
Likewise, for period2 motion, in which the motion repeats itself exactly every two periods of the
external drive, we expect the Poincaré section to consist of two points
in phasespace (i.e., we expect alternating points to plot on top of one another).
Finally, for period motion we expect the Poincaré section to consist of points
in phasespace.
Figure 63:
The Poincaré section of a timeasymptotic
orbit. Data calculated numerically for , , , ,
, and .

Figure 63 displays the Poincaré section of the orbit shown in Figure 62.
The fact that the section consists of a single point confirms that the motion
displayed in Figure 62 is indeed period1 motion.
Next: Spatial Symmetry Breaking
Up: The Chaotic Pendulum
Previous: Numerical Solution
Richard Fitzpatrick
20110331