Information Technology Reference
In-Depth Information
versus
Y
n
.
On the left the splatter of points seems to randomly fill the two-dimensional
space, strongly suggesting that the successive time points in the time series are truly
independent of one another. However, on the right we see that successive data points
fall on the curve that defines the map with which they were generated. The randomness
in the upper-right time series has to do with the order in which the values are generated
by the map; it is the order of appearance of the values of
Y
which looks random. All the
values arise from the curve on the lower right, but the values on this curve do not appear
in an ordered way but “randomly” jump from one value to the next. Chaos is the result
of the sensitivity to initial conditions, so, no matter how small a change one makes in
the initial condition, eventually the generated time series look totally different.
A coarse-grained version of this map, one in which values of
Y
in the interval
1
2are
assigned a second value, say 0, generates a Bernoulli sequence. Recall that the elements
of a Bernoulli sequence are statistically independent of one another. Thus, the logistic
map may be used as a random-number generator. The assumption that the sequence of
zeros and ones generated in this way form a Bernoulli sequence can be tested.
Another use of the logistic map is to identify
Y
n
with the following quantities:
in genetics, the change in the gene frequency between successive generations; in
epidemiology, the fraction of the population infected at a given time; in psychology, the
number of bits of information that can be remembered up to a generation
n
in certain
learning theories; in sociology, the number of people having heard a rumor at time
n
and how the rumor is propagated in societies of various structures; see May [
31
]for
a more extended discussion. Consequently, the potential application of these maps is
restricted only by our imaginations. Conrad [
16
] suggested five functional categories to
identify the role that chaos might play in biological networks: (1) search, (2) defense,
(3) maintenance, (4) cross-level effects and (5) dissipation of disturbances. But it must
also be acknowledged that using chaotic time series to model such phenomena has not
been universally embraced.
One of the measures of chaos is the attractor-reconstruction technique (ART) in
which the axes chosen for the phase space are empirical. These axes are intended to
provide an indication of the number of independent variables necessary to describe the
dynamical network. The idea is that a time series for a nonlinear dynamical process
contains information about all the variables necessary to describe the network. If the
number of variables is three then in the phase space the dynamics unfold on a three-
dimensional manifold. If the time series is fractal the three-dimensional Euclidean space
might contain a fractal manifold with fractal dimension in the interval 2
/
2
≤
Y
≤
1 are assigned one value, say +1, and those in the interval 0
≤
Y
≤
1
/
3. Even
in two dimensions the trajectory would be confined to a restricted region on which the
higher-dimensional dynamics would be projected.
An example of such an erratic time series is the electroencephalogram (EEG) of the
activity of the human brain. If
<
D
<
is a discretely sampled EEG record then a two-
dimensional phase space could be constructed from the data pairs
{
X
n
}
{
X
n
,
X
n
+
τ
}
, where
τ
is an empirically determined integer. In Figure
4.11
EEG time series recorded under
a variety of conditions are given on the left. On the right is the corresponding two-
dimensional phase space obtained using ART. While the phase-space structures lack
