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We have seen how the original fixed point X* bifurcates to give harmonics of period 2n. But how do new cycles of period k arise?

The general process is illustrated in Fig. 5, which shows how period 3 cycles originate. By an obvious extension of the notation introduced in equation (8). populations three generations apart are related by

Xt+3 = F(3) (Xt)         (11)

If the hump in F(X) is sufficiently steep, the threefold iteration will produce a function F(3)(X) with 4 humps, as shown in Fig. 5 for the F(X) of equation (3). At first (for a < 3.8284 . . in equation 3) the 45° line intersects this curve only at the single point X* (and at X = 0), as shown by the solid curve in Fig. 5. As the hump in F(X) steepens, the hills and valleys in F(3)(X) become more pronounced, until simultaneously the first two valleys sink and the final hill rises to touch the 45° line, and then to intercept it at 6 new points, as shown by the dashed curve in Fig. 5. These 6 points divide into two distinct three-point cycles. As can be made plausible by imagining the intermediate stages in Fig. 5, it can be shown that the stability-determining slope of F(3)(X) at three of these points has a common value, which is lambda(3) = +1 at their birth, and thereafter steepens beyond +1: this period 3 cycle is never stable. The slope of F(3)(X) at the other three points begins at lambda(3) = +1, and then decreases towards zero, resulting in a stable cycle of period 3. As F(X) continues to steepen, the slope lambda(3) for this initially stable three-point cycle decreases beyond -1; the cycle becomes unstable, and gives rise by the bifurcation process discussed in the previous section to stable cycles of period 6, 12, 24, ..., 3 × 2n. This birth of a stable and unstable pair of period 3 cycles, and the subsequent harmonics which arise as the initially stable cycle becomes unstable, are illustrated to the right of Fig. 4.

Figure 5

Figure 5. The relationship between Xt+3 and Xt, obtained by three iterations of equation (3). The solid curve is for a = 3.7, and only intersects the 45° line once. As a increases, the hills and valleys become more pronounced. The dashed curve is for a = 3.9, and six new period 3 points have appeared (arranged as two cycles, each of period 3).

There are, therefore, two basic kinds of bifurcation processes 1, 4 for first order difference equations. Truly new cycles of period k arise in pairs (one stable, one unstable) as the hills and valleys of higher iterates of F(X) move, respectively, up and down to intercept the 45° line, as typified by Fig. 5. Such cycles are born at the moment when the hills and valleys become tangent to the 45° line, and the initial slope of the curve F(k) at the points is thus lambda(k) = +1: this type of bifurcation may be called 1, 4 a tangent bifurcation or a lambda = +1 bifurcation. Conversely, an originally stable cycle of period k may become unstable as F(X) steepens. This happens when the slope of F(k) at these period k points steepens beyond lambda(k) = -1, whereupon a new and initially stable cycle of period 2k is born in the way typified by Figs 2 and 3. This type of bifurcation may be called a pitchfork bifurcation (borrowing an image from the left hand side of Fig. 4) or a lambda = -1 bifurcation 1, 4.

Putting all this together, we conclude that as the parameters in F(X) are varied the fundamental, stable dynamical units are cycles of basic period k, which arise by tangent bifurcation, along with their associated cascade of harmonics of periods k2n, which arise by pitchfork bifurcation. On this basis, the constant equilibrium solution X* and the subsequent hierarchy of stable cycles of periods 2n is merely a special case, albeit a conspicuously important one (namely k = 1), of a general phenomenon. In addition, remember 1, 4, 22, 29 that for sensible, analytical functions (such as, for example, those in equations (3) and (4)) there is a unique stable cycle for each value of the parameter in F(X). The entire range of parameter values (1 < a < 4 in equation (3), 0 < r in equation (4)) may thus be regarded as made up of infinitely many windows of parameter values - some large, some unimaginably small - each corresponding to a single one of these basic dynamical units. Tables 3 and 4, below, illustrate this notion. These windows are divided from each other by points (the points of accumulation of the harmonics of period k2n) at which the system is truly chaotic, with no attractive cycle: although there are infinitely many such special parameter values, they have measure zero on the interval of all values.

How are these various cycles arranged along the interval of relevant parameter values? This question has to my knowledge been answered independently by at least 6 groups of people, who have seen the problem in the context of combinatorial theory 16, 30, numerical analysis13, 14, population biology 1, and dynamical systems theory 22, 31 (broadly defined).

A simple-minded approach (which has the advantage of requiring little technical apparatus, and the disadvantage of being rather clumsy) consists of first answering the question, how many period k points can there be? That is, how many distinct solutions can there be to the equation

X*k = F(k) (X*k)?         (12)

If the function F(X) is sufficiently steeply humped, as it will be once the parameter values are sufficiently large, each successive iteration doubles the number of humps, so that F(k)(X) has 2k-1 humps. For large enough parameter values, all these hills and valleys will intersect the 45° line, producing 2k fixed points of period k. These are listed for k leq 12 in the top row of Table 2. Such a list includes degenerate points of period k, whose period is a submultiple of k; in particular, the two period 1 points (X = 0 and X*) are degenerate solutions of equation (12) for all k. By working from left to right across Table 2, these degenerate points can be subtracted out, to leave the total number of non-degenerate points of basic period k, as listed in the second row of Table 2. More sophisticated ways of arriving at this result are given elsewhere 13, 14, 16, 22, 30, 31.

Table 2. Catalogue of the number of periodic points, and of the various cycles (with periods k = 1 up to 12), arising from equation (1) with a single-humped function F(X)

k 1 2 3 4 5 6 7 8 9 10 11 12
Possible total number of points with period k 2 4 8 16 32 64 128 256 5121,024 2,048 4,096
Possible total number of points with non-degenerate period k 2 2 6 12 30 54 126 240 504 990 2,046 4,020
Total number of cycles of period k, including those which are degenerate and/or harmonics and/or never locally stable 2 3 4 6 8 14 20 36 60 108 188 352
Total number of non-degenerate cycles (including harmonics and unstable cycles) 2 1 2 3 6 9 18 30 56 99 186 335
Total number of non-degenerate, stable cycles (including harmonics) 1 1 1 2 3 5 9 16 28 51 93 170
Total number of non-degenerate, stable cycles whose basic period is k (that is, excluding harmonics) 1 - 1 1 3 4 9 14 28 48 93 165

For example, there eventually are 26 = 64 points with period 6. These include the two points of period 1, the period 2 "harmonic" cycle, and the stable and unstable pair of triplets of points with period 3, for a total of 10 points whose basic period is a submultiple of 6; this leaves 54 points whose basic period is 6.

The 2k period k points are arranged into various cycles of period k, or submultiples thereof, which appear in succession by either tangent or pitchfork bifurcation as the parameters in F(X) are varied. The third row in Table 2 catalogues the total number of distinct cycles of period k which so appear. In the fourth row 14, the degenerate cycles are subtracted out, to give the total number of non-degenerate cycles of period k: these numbers must equal those of the second row divided by k. This fourth row includes the (stable) harmonics which arise by pitchfork bifurcation, and the pairs of stable-unstable cycles arising by tangent bifurcation. By subtracting out the cycles which are unstable from birth, the total number of possible stable cycles is given in row five; these figures can also be obtained by less pedestrian methods 13, 16, 30. Finally we may subtract out the stable cycles which arise by pitchfork bifurcation, as harmonics of some simpler cycle, to arrive at the final row in Table 2, which lists the number of stable cycles whose basic period is k.

Returning to the example of period 6, we have already noted the five degenerate cycles whose periods are submultiples of 6. The remaining 54 points are parcelled out into one cycle of period 6 which arises as the harmonic of the only stable three-point cycle, and four distinct pairs of period 6 cycles (that is, four initially stable ones and four unstable ones) which arise by successive tangent bifurcations. Thus, reading from the foot of the column for period 6 in Table 2, we get the numbers 4, 5, 9, 14.

Using various labelling tricks, or techniques from combinatorial theory, it is also possible to give a generic list of the order in which the various cycles appear 1, 13, 16, 22. For example, the basic stable cycles of periods 3, 5, 6 (of which there are respectively 1, 3, 4) must appear in the order 6, 5, 3, 5, 6, 6, 5, 6: compare Tables 3 and 4. Metropolis et al. 16 give the explicit such generic list for all cycles of period k leq 11.

Table 3 A catalogue of the stable cycles (with basic periods up to 6) for the equation Xt+1 = aXt(1 - Xt)

Width of the range
a value at which: Subsequent cascade of a values over
of "harmonics" with which the basic cycle,
Period of Basic cycle Basic cycle period k2n all or one of its harmonics,
basic cycle first appears becomes unstable become unstable is attractive
1 1.0000 3.0000 3.5700 2.5700
3 3.8284 3.8415 3.8495 0.0211
4 3.9601 3.9608 3.9612 0.0011
5(a) 3.7382 3.7411 3.7430 0.0048
5(b) 3.9056 3.9061 3.9065 0.0009
5(c) 3.99026 3.99030 3.99032 0.00006
6(a) 3.6265 3.6304 3.6327 0.0062
6(b) 3.937516 3.937596 3.937649 0.000133
6(c) 3.977760 3.977784 3.977800 0.000040
6(d) 3.997583 3.997585 3.997586 0.000003

As a corollary it follows that, given the most recent cycle to appear, it is possible (at least in principle) to catalogue all the cycles which have appeared up to this point. An especially elegant way of doing this is given by Smale and Williams 22, who show, for example, that when the stable cycle of period 3 first originates, the total number of other points with periods k, Nk, which have appeared by this stage satisfy the Fibonacci series, Nk = 2, 4, 5, 8, 12, 19, 30, 48, 77, 124, 200, 323 for k = 1, 2, ..., 12: this is to be contrasted with the total number of points of period k which will eventually appear (the top row of Table 2) as F(X) continues to steepen.

Table 4. Catalogue of the stable cycles (with basic periods up to 6) for the equation Xt+1 = Xt exp[r(1 - Xt)]

Width of the range
r value at which: Subsequent cascade of r values over
of "harmonics" with which the basic cycle,
Period of Basic cycle Basic cycle period k2n all or one of its harmonics,
basic cycle first appears becomes unstable become unstable is attractive
1 0.0000 2.00002.6924 2.6924
3 3.1024 3.15963.1957 0.0933
4 3.5855 3.60433.6153 0.0298
5(a)2.9161 2.92222.9256 0.0095
5(b)3.3632 3.36643.3682 0.0050
5(c)3.9206 3.92953.9347 0.0141
6(a)2.7714 2.77612.7789 0.0075
6(b)3.4558 3.45633.4567 0.0009
6(c)3.7736 3.77453.7750 0.0014
6(d)4.1797 4.18484.1880 0.0083

Such catalogues of the total number of fixed points, and of their order of appearance, are relatively easy to construct. For any particular function F(X), the numerical task of finding the windows of parameter values wherein any one cycle or its harmonics is stable is, in contrast, relatively tedious and inelegant. Before giving such results, two critical parameter values of special significance should be mentioned.

Hoppensteadt and Hyman 21 have given a simple graphical method for locating the parameter value in the chaotic regime at which the first odd period cycle appears. Their analytic recipe is as follows. Let alpha be the parameter which tunes the steepness of F(X) (for example, alpha = a for equation (3), alpha = r for equation (4)), X*(alpha) be the fixed point of period 1 (the non-trivial solution of equation (5)), and Xmax(alpha) the maximum value attainable from iterations of equation (1) (that is, the value of F(X) at its hump or stationary point). The first odd period cycle appears for that value of alpha which satisfies 21, 31

X*(alpha) = F(2) (Xmax (alpha))         (13)

As mentioned above, another critical value is that where the period 3 cycle first appears. This parameter value may be found numerically from the solutions of the third iterate of equation (1): for equation (3) it is 14 a = 1 + sqrt8.

Myrberg 13 (for all k leq 10) and Metropolis et al. 16. (for all k leq 7) have given numerical information about the stable cycles in equation (3). They do not give the windows of parameter values, but only the single value at which a given cycle is maximally stable; that is, the value of a for which the stability-determining slope of F(k)(X) is zero, lambda(k) = 0. Since the slope of the k-times iterated map F(k) at any point on a period k cycle is simply equal to the product of the slopes of F(X) at each of the points X*k on this cycle 1, 8, 20, the requirement lambda(k) = 0 implies that X = A (the stationary point of F(X), where lambda(1) = 0) is one of the periodic points in question, which considerably simplifies the numerical calculations.

For each basic cycle of period k (as catalogued in the last row of Table 2), it is more interesting to know the parameter values at which: (1) the cycle first appears (by tangent bifurcation); (2) the basic cycle becomes unstable (giving rise by successive pitchfork bifurcations to a cascade of harmonics of periods k2n); (3) all the harmonics become unstable (the point of accumulation of the period k2n cycles). Tables 3 and 4 extend the work of May and Oster1, to give this numerical information for equations (3) and (4), respectively. (The points of accumulation are not ground out mindlessly, but are calculated by a rapidly convergent iterative procedure, see ref. 1, appendix A.) Some of these results have also been obtained by Gumowski and Mira 32.

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