Information Technology Reference
In-Depth Information
the discussion of dynamics with maps and see how even the simplest nonlinear map has
chaotic solutions.
In previous chapters the dynamics of the variables thought to describe complex webs
were addressed from a variety of perspectives. We examined the conservative dynamics
of classical mechanics and Hamilton's equations of motion. In this chapter we show
how linear interactions give way to nonlinear interactions resulting in the dynamics
of chaos as web interactions become more complicated. In this replacement of the
linear with the nonlinear it also becomes clear that the precise and certain predictions
for which scientists strive are not available when the web is truly complex. Chaos,
with its sensitive dependence on initial conditions, provides predictions for only very
short times, since infinitesimally distant trajectories exponentially separate from one
another in time. So the chaos of unstable Hamiltonian trajectories, as well as the
strange attractors of dissipative networks, both lead to complexity that defies long-time
predictability in real webs.
Part of our purpose is to seek out and/or develop analysis techniques that are not
restricted to describing the behavior of physical webs. More specifically we are looking
for universal principles such as the variational principle used in the Hamiltonian formal-
ism; that is, the variation of the total energy vanishes and yields the equations of motion.
Here we find that linear response theory (LRT) in conjunction with chaos theory pro-
vides a dynamical foundation for the fluctuation-dissipation relation. This marriage of
the two concepts suggests that LRT may be universal; this is because it does not refer to
a single trajectory, but rather to the manifold in phase space on which all the trajector-
ies in an ensemble of initial conditions unfold. Thus, although the manifold is linearly
distorted by a perturbation, the underlying dynamics of the process can be arbitrarily
deformed and consequently the dynamic response of the web is not linear. However,
unlike the response of a single trajectory, the average dynamical response of the web to
the perturbation
is
linear. The average is taken over the chaotic fluctuations generated
by the microscopic variables.
A Hamiltonian coupling a macroscopic variable to microscopic chaos is used to
motivate the universality of LRT to model the average web response to perturbation.
This argument sets the stage for the more rigorous presentation in later chapters.
But for the time being the formalism establishes computational consistency between
a microscopic nonlinear dynamic model coupled to a macroscopic variable and
Brownian motion; microscopic chaos is manifest as fluctuations in the equation for the
macroscopic variable.
4.1
Random walks
The disruption of web predictability was investigated at the turn of the last century by
introducing random forces into dynamical equations. In physics these stochastic differ-
ential equations are called Langevin equations for historical reasons and were discussed
in the previous chapter. In Langevin equations the random force is intended to mimic
the influence of the environment on the dynamics of the web of interest. In this way
