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which cycles through the same values periodically. The time it takes each oscillator to
go through a cycle is different and depends on the oscillator frequency
τ
j
=
2
π/ω
j
,the
oscillator period.
3.2
Linear stochastic equations
One strategy for developing an understanding of complex webs is to start from the
simplest dynamical network that we have previously analyzed and embellish it with
increasingly more complicated mechanisms. The linear harmonic oscillator is the usual
starting point and it can be extended by introducing an external driver, a driver repre-
sented by a function that can be periodic, random or completely general. You are asked
to examine the response of a linear harmonic oscillator to a periodic driver in Problem
3.1. But now let us turn our attention to the response of a linear web to the influence
of random forces. Here we continue to follow the physics approach and partition the
universe into the web of interest and the environment (everything else).
We cannot know the physical properties of the environment in a finite experiment
because the environment can be indefinitely complex. As a result of this unknowability
the environment is not under our control in any given experiment, only knowledge of
the much smaller-dimensional web is accessible to us. Consequently, the environment's
influence on the experimental web is unpredictable and unknowable except in an aver-
age way when the experiment is repeated again and again. It is this repeatability of the
experiment that provides the data showing the different ways the environment influ-
ences the network. Functionally it is the ensemble distribution function that captures
the patterns common to all the experiments in the ensemble, as well as the degree of
difference from one experiment to the next.
One way we can model this uncertainty effect is by assuming that the linear oscillators
of the last section are pushed and pulled in unknown ways by the environment before
we begin to track their dynamics. Consequently, we assume that the initial phase of
each oscillator
φ
j
is uniformly distributed on the interval 0
≤
φ
j
≤
2
π
and therefore a
function of the total oscillator displacements,
N
F
(
t
)
=
A
j
cos
[
ω
j
t
+
φ
j
]
,
(3.38)
j
=
1
is random. This was one of the first successful models to describe the dynamical dis-
placement of a point on the surface of the deep ocean [
31
] in the middle of the last
century. Here
A
j
is the amplitude of a water wave,
ω
j
is its frequency and the displace-
ment of the ocean's surface at a point in space (
3.38
) is the linear superposition of these
waves, each with a phase that is shifted randomly with respect to one another,
This
is sometimes called the random-phase approximation. This function was apparently first
used by Lord Rayleigh at the end of the nineteenth century to represent the total sound
received from
N
incoherent point sources [
40
]. Note that this function
F
φ
j
.
consists
of a large number of identically distributed random variables, so that if the variance of
(
t
)
