We have seen how the original fixed point *X** bifurcates to give
harmonics of period 2^{n}. 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

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
^{(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
^{(3)} = +1, and
then decreases
towards zero, resulting in a stable cycle of period 3. As *F(X)*
continues to steepen, the slope
^{(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 × 2^{n}. 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.

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
^{(k)} = +1: this
type of bifurcation may be called
^{1,
4}
a tangent bifurcation or a
= +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
^{(k)} = -1, whereupon
a new and initially stable cycle of period 2*k* 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
= -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
*k*2^{n}, which arise by pitchfork bifurcation. On this
basis, the
constant equilibrium solution *X** and the subsequent hierarchy
of stable cycles of periods 2^{n} 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 *k*2^{n}) 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 analysis^{13,
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

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
2^{k-1} humps. For large enough parameter values, all these hills
and valleys will intersect the 45° line, producing 2^{k}
fixed points
of period *k*. These are listed for *k*
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}.

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 | 512 | 1,024 | 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 2^{6} = 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 2^{k} 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
11.

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 k2^{n} 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, N*_{k}, which have appeared by this stage satisfy the
Fibonacci
series, *N*_{k} = 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.

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 k2^{n} all | or one of its harmonics, |

basic cycle | first appears | becomes unstable | become unstable | is attractive |

1 | 0.0000 | 2.0000 | 2.6924 | 2.6924 |

3 | 3.1024 | 3.1596 | 3.1957 | 0.0933 |

4 | 3.5855 | 3.6043 | 3.6153 | 0.0298 |

5(a) | 2.9161 | 2.9222 | 2.9256 | 0.0095 |

5(b) | 3.3632 | 3.3664 | 3.3682 | 0.0050 |

5(c) | 3.9206 | 3.9295 | 3.9347 | 0.0141 |

6(a) | 2.7714 | 2.7761 | 2.7789 | 0.0075 |

6(b) | 3.4558 | 3.4563 | 3.4567 | 0.0009 |

6(c) | 3.7736 | 3.7745 | 3.7750 | 0.0014 |

6(d) | 4.1797 | 4.1848 | 4.1880 | 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
be the parameter which tunes the
steepness of *F(X)* (for example,
= *a* for equation
(3), = *r* for
equation (4)), *X**()
be the fixed point of period 1 (the
non-trivial solution of equation (5)), and
*X*_{max}() 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
which satisfies
^{21,
31}

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 +
8.

Myrberg ^{13}
(for all k 10) and Metropolis
et al.
^{16}. (for all
k 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,
^{(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
^{(k)} = 0
implies that *X* = A (the stationary point of *F(X)*, where
^{(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 *k*2^{n}); (3) all the harmonics become unstable
(the point
of accumulation of the period *k*2^{n} cycles).
Tables 3 and 4 extend
the work of May and Oster^{1}, 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}.