As discussed above, the essential reason for the existence of a succession of stable cycles throughout the "chaotic" regime is that as each new pair of cycles is born by tangent bifurcation (see Fig. 5), one of them is at first stable, by virtue of the way the smoothly rounded hills and valleys intercept the 45° line. For analytical functions F(X), the only parameter values for which the density plot or "invariant measure" is continuous and truly ergodic are at the points of accumulation of harmonics, which divide one stable cycle from the next. Such exceptional parameter values have found applications, for example, in the use of equation (3) with a = 4 as a random number generator 34, 35: it has a continuous density function proportional to [X(1 - X)]-1/2 in the interval 0 < X < 1.
Non-analytical functions F(X) in which the hump is in fact a
spike provide an interesting special case. Here we may imagine
spikey hills and valleys moving to intercept the 45° line in
Fig. 5,
and it may be that both the cycles born by tangent bifurcation
are unstable from the outset (one having
(k) > 1, the
other (k) < -1),
for all k > 1. There are then no stable cycles in the
chaotic regime, which is therefore literally chaotic with a
continuous and truly ergodic density distribution function.
One simple example is provided by
| Xt+1 = a Xt ; if Xt < 1/2 |
| Xt+1 = a(1 - Xt) ; if Xt > 1/2 (14) |
defined on the interval 0 < X < 1. For 0 < a <
1, all trajectories
are attracted to X = 0; for 1 < a < 2, there are
infinitely many
periodic orbits, along with an uncountable number of aperiodic
trajectories, none of which are locally stable. The first odd
period cycle appears at a =
2, and all integer periods are
represented beyond a = (1 +
5)/2. Kac36 has
given a careful
discussion of the case a = 2. Another example, this time with an
extensive biological pedigree
1 -
3, is the equation
| Xt+1 =
Xt ; if Xt < 1
|
| Xt+1 =
Xt1-b ; if Xt > 1
(15)
|
If > 1 this possesses
a globally stable equilibrium point for
b < 2. For b > 2 there is again true chaos, with no
stable cycles:
the first odd cycle appears at b = (3 +
5)/2, and all integer
periods are present beyond b = 3. The dynamical properties of
equations (14) and (15) are summarised to the right of
Table 2.
The absence of analyticity is a necessary, but not a sufficient, condition for truly random behaviour 31. Consider, for example,
| Xt+1 = (a / 2) Xt ; if Xt < ½ |
| Xt+1 = a Xt (1 - Xt) ; if Xt > ½ (16) |
This is the parabola of equation (3) and
Fig. 1, but with the
left hand half of F(X) flattened into a straight line. This equation
does possess windows of a values, each with its own stable
cycle, as described generically above. The stability-determining
slopes (k)
vary, however, discontinuously with the parameter a,
and the widths of the simpler stable regions are narrower than
for equation (3): the fixed point becomes unstable at a = 3; the
point of accumulation of the subsequent harmonics is at
a = 3.27 . .; the first odd cycle appears at a = 3.44 . .; the
3-point cycle at a = 3.67. . (compare the first column in
Table 1).
These eccentricities of behaviour manifested by non-analytical functions may be of interest for exploring formal questions in ergodic theory. I think, however, that they have no relevance to models in the biological and social sciences, where functions such as F(X) should be analytical. This view is elaborated elsewhere 37.
As a final curiosity, consider the equation
Xt [1 +
Xt]-
(17)
This has been used to fit a considerable amount of data on
insect populations 38,
39.
Its stability behaviour, as a function of
the two parameters and
, is illustrated in
Fig. 6. Notice that
for < 7.39 . . there
is a globally stable equilibrium point for
all ; for 7.39. . <
X < 12.50. . this fixed point becomes unstable
for sufficiently large ,
bifurcating to a stable 2-point cycle
which is the solution for all larger
; as increases through the
range 12.50 . . < <
14.77 . . various other harmonics of
period 2n appear in turn. The hierarchy of bifurcating cycles of
period 2n is thus truncated, and the point of accumulation and
subsequent regime of chaos is not achieved (even for arbitrarily
large ) until
> 14.77...
|
Figure 6. The solid lines demarcate the
stability domains for the
density dependence parameter,
|