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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 lambda(k) > 1, the other lambda(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 = sqrt2, and all integer periods are represented beyond a = (1 + sqrt5)/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 = lambda Xt ; if Xt < 1
Xt+1 = lambda Xt1-b ; if Xt > 1         (15)

If lambda > 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 + sqrt5)/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 lambda(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 = lambda Xt [1 + Xt]-beta         (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 lambda and beta, is illustrated in Fig. 6. Notice that for lambda < 7.39 . . there is a globally stable equilibrium point for all beta; for 7.39. . < X < 12.50. . this fixed point becomes unstable for sufficiently large beta, bifurcating to a stable 2-point cycle which is the solution for all larger beta; as increases through the range 12.50 . . < lambda < 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 beta) until lambda > 14.77...

Figure 6

Figure 6. The solid lines demarcate the stability domains for the density dependence parameter, beta, and the population growth rate, lambda, in equation (17); the dashed line shows where 2-point cycles give way to higher cycles of period 2n. The solid circles come from analyses of life table data on field populations, and the open circles from laboratory populations (from ref. 3, after ref. 39).

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