The route to chaos

Figure 44 is a blow-up of Fig. 43, showing more details of the onset of
chaos. The period-4 to period-8 bifurcation can be seen quite clearly. However,
we can also see a period-8 to period-16 bifurcation, at
. Finally,
if we look carefully, we can see a hint of a period-16 to period-32 bifurcation, just
before the start of the solid black region.
Figures 43 and 44 seem to suggest that the
onset of chaos is triggered by an *infinite series* of period-doubling bifurcations.

Table 2 gives some details of the sequence of period-doubling bifurcations
shown in Figs. 43 and 44. Let us introduce a bifurcation index : the
period-1 to period-2 bifurcation corresponds to ; the
period-2 to period-4 bifurcation corresponds to ; and so on. Let be
the critical value of the quality-factor above which the th bifurcation is
triggered. Table 2 shows the , determined from Figs. 43 and 44, for
to 5. Also shown is the ratio

(98) |

(99) |

(100) |

Let us examine the onset of chaos in a little more detail. Figures 45-48
show details of the pendulum's time-asymptotic motion at various stages on the
period-doubling cascade discussed above. Figure 45 shows period-4 motion:
note that the Poincaré section consists of four points, and the associated sequence of
net rotations per period of the pendulum repeats itself every four periods.
Figure 46 shows period-8 motion:
now the Poincaré section consists of eight points, and the rotation sequence
repeats itself every eight periods. Figure 47 shows period-16 motion:
as expected, the Poincaré section consists of sixteen points, and the rotation sequence
repeats itself every sixteen periods. Finally, Fig. 48 shows chaotic motion. Note that
the Poincaré section now consists of a set of four *continuous* line segments, which are, presumably, made
up of an infinite number of points (corresponding to the infinite period of chaotic motion).
Note, also, that the associated sequence of net rotations per period shows no obvious sign of ever
repeating itself. In fact, this sequence looks rather like one of the previously shown periodic
sequences with the addition of a small random component. The generation of *apparently
random* motion from equations of motion, such as Eqs. (81) and
(82), which contain no overtly random elements is one of the most surprising features of
non-linear dynamics.

Many non-linear dynamical systems, found in nature, exhibit a transition
from periodic to chaotic motion as some control parameter is varied.
Now, there are various known mechanisms by which chaotic motion can arise from periodic
motion. However, a transition to chaos via an infinite series of period-doubling bifurcations, as
illustrated
above, is certainly amongst the most commonly occurring of these mechanisms.
Around 1975, the physicist Mitchell Feigenbaum was investigating a simple mathematical
model, known as the *logistic map*, which exhibits a transition to chaos,
via a sequence of period-doubling bifurcations, as a control parameter
is increased. Let be the value of at which the
first -period cycle appears. Feigenbaum noticed that the ratio

(101) |

The existence of a universal ratio characterizing the transition to chaos via
period-doubling bifurcations is one of many pieces of evidence indicating that
chaos is a *universal* phenomenon (*i.e.*, the onset and nature
of chaotic motion in different dynamical systems has many common features).
This observation encourages us to believe that in studying the chaotic motion of
a damped, periodically driven, pendulum we are learning lessons which can
be applied to a wide range of non-linear dynamical systems.