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(a)
(b)
0.8
0.7
0.7
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0 0 0
Generation
n
30
40
50
0 0 0
Generation
n
30
40
50
(c)
(d)
0.8
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.6
0.4
0.2
0 0 0
Generation
n
30
40
50
0 0 0
Generation
n
30
40
50
Figure 4.9.
The solution to the discrete logistic equation is given as a function of iteration number
n
for
values of the control parameter: (a)
λ
=
2
.
9 in the region of a single asymptotic state; (b)
λ
=
3
.
2 yields a 2-cycle solution; (c)
λ
=
3
.
5 yields a region with 4-cycle solutions; and (d)
λ
=
.
3
9 is a region of bounded chaos.
In Figure
4.9
(c) the value of the control parameter is
This value of the control
parameter gives rise to two bifurcations and a solution that oscillates among four val-
ues, asymptotically a 4-cycle. Finally, in Figure
4.9
(d) the control parameter
λ
=
3
.
5
.
λ
=
.
9is
selected. In this last case the dynamics are chaotic, which is to say that there is appar-
ently a complete loss of regularity in the solution. This would appear to be complex
dynamical behavior.
Thus, using a traditional definition of complexity, it would appear that chaos implies
the generation of complexity from simplicity. It is also worth noting that chaos is
a generic property of nonlinear dynamical webs, that is, it is ubiquitous; all net-
works change over time, and, because all naturally occurring dynamical networks are
nonlinear, all such webs manifest chaotic behavior to a greater or lesser extent.
One way to interpret what is being represented by maps, such as the logistic equa-
tion, is that the time points in the map are equally spaced values of a continuous process.
Consider a dance floor that is lit by a stroboscopic lamp, otherwise the dancers are in
complete darkness. The motion of the dancers seems to occur in abrupt moves, but
you can deduce where the dancers were during the periods of darkness because human
motion is continuous. In the same way the dynamical unfolding of a continuous process
can be recorded at a set of equally spaced intervals, say every second, or at unequally
spaced time intervals, say by recording the amplitude each time the derivative of the
time series vanishes, or using any of a number of other criteria to discretize continuous
time series. The data points are related because they are each generated by the same
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