Referring to the paradigmatic example of equation (3), we can
now see that the parameter interval 1 < *a* < 4 is made up of a
one-dimensional mosaic of infinitely many windows of *a*-values,
in each of which a unique cycle of period *k*, or one of its
harmonics, attracts essentially all initial points. Of these
windows, that for 1 < *a* < 3.5700 . . corresponding to
*k* = 1 and
its harmonics is by far the widest and most conspicuous. Beyond
the first point of accumulation, it can be seen from
Table 3 that
these windows are narrow, even for cycles of quite low periods,
and the windows rapidly become very tiny as *k* increases.

As a result, there develops a dichotomy between the underlying
mathematical behaviour (which is exactly determinable)
and the "commonsense" conclusions that one would draw from
numerical simulations. If the parameter *a* is held constant at one
value in the chaotic region, and equation (3) iterated for an
arbitrarily large number of generations, a density plot of the
observed values of *X*_{t} on the interval 0 to 1 will
settle into *k*
equal spikes (more precisely, delta functions) corresponding to
the *k* points on the stable cycle appropriate to this
*a*-value. But
for most *a*-values this cycle will have a fairly large period, and
moreover it will typically take many thousands of generations
before the transients associated with the initial conditions are
damped out: thus the density plot produced by numerical
simulations usually looks like a sample of points taken from
some continuous distribution.

An especially interesting set of numerical computations are
due to Hoppensteadt (personal communication) who has
combined many iterations to produce a density plot of
*X*_{t} for
each one of a sequence of *a*-values, gradually increasing from
3.5700 . . to 4. These results are displayed as a movie. As can
be expected from Table 3, some of the
more conspicuous cycles
do show up as sets of delta functions: the 3-cycle and its first
few harmonics; the first 5-cycle; the first 6-cycle. But for most
values of *a* the density plot looks like the sample function of a
random process. This is particularly true in the neighbourhood
of the *a*-value where the first odd cycle appears (*a* =
3.6786..),
and again in the neighbourhood of *a* = 4: this is not surprising,
because each of these locations is a point of accumulation of
points of accumulation. Despite the underlying discontinuous
changes in the periodicities of the stable cycles, the observed
density pattern tends to vary smoothly. For example, as *a*
increases toward the value at which the 3-cycle appears, the
density plot tends to concentrate around three points, and it
smoothly diffuses away from these three points after the 3-cycle
and all its harmonics become unstable.

I think the most interesting mathematical problem lies in
designing a way to construct some approximate and "effectively
continuous" density spectrum, despite the fact that the exact
density function is determinable and is always a set of delta
functions. Perhaps such techniques have already been developed
in ergodic theory
^{33} (which lies
at the foundations of
statistical mechanics), as for example in the use of "coarse-grained
observers". I do not know.

Such an effectively stochastic description of the dynamical
properties of equation (4) for large *r* has been provided
^{28}, albeit
by tactical tricks peculiar to that equation rather than by any
general method. As *r* increases beyond about 3, the trajectories
generated by this equation are, to an increasingly good approximation,
almost periodic with period (1 / *r*) exp(*r* - 1).

The opinion I am airing in this section is that although the
exquisite fine structure of the chaotic regime is mathematically
fascinating, it is irrelevant for most practical purposes. What
seems called for is some effectively stochastic description of the
deterministic dynamics. Whereas the various statements about
the different cycles and their order of appearance can be made
in generic fashion, such stochastic description of the actual
dynamics will be quite different for different *F(X)*: witness the
difference between the behaviour of equation (4), which for
large *r* is almost periodic "outbreaks" spaced many generations
apart, versus the behaviour of equation (3), which for *a* -> 4 is
not very different from a series of Bernoulli coin flips.