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the FRF/USD and the DM/USD time series because the Hurst index in these cases
approaches the deterministic value of one.
Anomalous diffusion
If we restrict the discussion to random walks then a second moment of a time series that
increases algebraically in time,
Q
2
)
∝
t
α
, describes a phenomenon that can poten-
tially be separated into three parts: classical diffusion with
(
t
α
=
1, superdiffusion with
2
≥
α>
1 and subdiffusion with 1
≥
α>
0. Any algebraic time behavior other than
α
=
1 is referred to as anomalous. In the simple random-walk context it is assumed in
the literature that
.
However, in the above discussion a random force with long-term memory was shown
to induce a power-law exponent
α
=
2
H
and the Hurst exponent is restricted to the range 1
≥
H
>
0
1, a value not allowed in the
simple random-walk picture but which is reasonable when memory is included in the
description. Consequently, such general random walks might also be used to “explain”
the inverse power laws observed in the connections between elements in certain kinds of
complex webs. The question of whether this modeling of the second moment is unique
arises and, if it is not, is there a test for the non-uniqueness?
The interpretation in terms of long-time memory is a consequence of the ordering of
the data in the time series. If the data points in the time series are randomly shuffled the
memory disappears; however, since no data points are added or subtracted in the shuf-
fling process, the statistics of the time series remains unchanged. A direct test of whether
the algebraic form of the second moment results from memory or from some exotic sta-
tistical distribution is to randomly rearrange the data points and recalculate the second
moment. In this way a time series with a second moment with
α>
2 or equivalently
H
>
α
=
1 before shuffling
and
1 after shuffling is shown to have long-time memory. On the other hand, if the
value of the power-law index is not changed by shuffling the data, the anomaly in the
second moment is due to non-Gaussian statistics.
We subsequently determine that the observed scaling of the second moment can
guide us in the formulation of the proper model of the 1
α
=
f
phenomenon. However,
phenomenological random walks are not the only way in which statistics can be intro-
duced into the dynamics of a process. Another way in which the variability can appear
random is through the chaotic solutions of nonlinear equations of motion, as mentioned
earlier. So now let us turn our attention to maps, chaos and strange attractors.
/
4.3
Maps, chaos and strange attractors
As late as the end of the 1960s, when the authors were graduate students in physics,
it was not general knowledge that the traditional techniques for solving the equations
of motion for complex mechanical networks were fundamentally flawed. Most physi-
cists were ignorant of the defects in the canonical perturbation methods developed for
celestial mechanics. These techniques were taught by them unabashedly in most grad-
uate courses on analytic dynamics. The story of the documentation of the breakdown
of canonical perturbation theory begins in 1887 when the king of Sweden, Oscar II,
