Information Technology Reference
In-Depth Information
that the act of measurement itself introduces a small and uncontrollable error into the
quantity being measured. Unlike the law of errors conceived by Gauss, which is based
on linearity and the principle of superposition of independent events, the postulated
errors arising from nonlinearities cannot be reduced by increasing the accuracy of one's
measurements. The new kind of error is generated by the intrinsic chaos associated with
physical being. Therefore, unlike the bell-shaped distribution resulting from the lin-
ear additive error concepts of Gauss, which have inappropriately been applied to many
social, medical and psychological webs, we have inverse power-law distributions that
require the underlying process to be contingent rather than independent, multiplicative
rather than additive and nonlinear rather than linear.
4.3.2
Strange attractors
Most phenomena of interest cannot be described by a single variable, so even when we
suspect that the dynamics being examined are chaotic, those dynamics are probably not
described by the one-humped maps just discussed. It is often the case that we do not
know how many variables are necessary to describe the web of interest. For example,
in something as complex as the human brain we are restricted by what we can measure,
not by what we know to be important. Thus, the EEG signal is often used to characterize
the operation of the brain even though we do not know how this voltage trace recorded
in the various EEG channels is related to brain function. This is a familiar situation to
the experimenter who is attempting to understand the behavior of a complex web. A few
variables are measured and from these data sets (usually time series) the fundamental
properties of the web must be induced. Such induction is often made irrespective of
whether or not the web is modeled by a set of deterministic dynamical equations. The
difficulty is compounded when the time series is erratic because then one must deter-
mine whether the random variations are due to noise in the traditional sense of the web
interacting with the environment, or whether the fluctuations are due to chaos, which is
an implicit dynamical property of the web.
Now we bring together the ideas of a chaotic map and the continuous dynamical
equations of mechanics. Envision the one-dimensional mapping as being the intersec-
tion of a continuous trajectory with a line. If the map is chaotic, this implies that the
original continuous trajectory is also chaotic. To generalize this remark to higher-order
networks, consider the intersection of a higher-order dynamical process with a two-
dimensional plane. A trajectory in three dimensions is sketched in Figure
4.12
and its
intersection with a cross-hatched plane is indicated. For a Hamiltonian web the trajec-
tory is a solution to the equations of motion and lies on the surface of a torus. However,
we are interested only in the discrete times when this trajectory intersects the plane, the
so-called Poincaré surface of section. The points
{
X
n
,
Y
n
}
denote the intersections of
the trajectory with the orthogonal axes
x
and
y
on the plane. These points can also be
generated by the two-dimensional map
X
n
+
1
=
g
1
(
X
n
,
Y
n
),
(4.81)
Y
n
+
1
=
g
2
(
X
n
,
Y
n
).
