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the mechanical behavior of physical networks, to the generic dynamics of other complex
webs. For the moment we quote Poincaré's great theorem:
The canonical equations of celestial mechanics do not admit (except for some exceptional
cases ...) any analytic and uniform integral besides the energy integral.
Of course, deterministic equations are only one way to describe how a complex web
changes over time and chaos is only one form of uncertainty. But it is worth pointing out
that there is more than one kind of chaos. One type of chaos that is not widely known
outside physics is associated with the breakdown of nonlinear conservative dynamic
webs. Another sort of chaos has to do with strange attractors, those being nonlinear
webs with dissipation. We only briefly touch on the latter here, but enough to trace
the source of unpredictability to the dynamics. The other source of unpredictability has
to do with the statistics of webs arising from their contact with infinitely unknowable
environments. Linear webs of this kind are modeled by linear stochastic differential
equations as we saw in the last chapter, and those whose complexity is influenced by
memory of the past are accounted for using fractional stochastic differential equations
as discussed in the previous section.
The oldest scientific discipline dealing with dynamics is classical mechanics dat-
ing from the attempts of Galileo and Newton to describe the dynamics of our physical
world. Newton's formalism developed into the equations of motion of Lagrange and
then into those of Hamilton. In the middle of the last century it became well known
that classical mechanics suffered from certain fundamental flaws and the theory of
Kolmogorov [
27
], Arnol'd [
3
] and Moser [
34
] (KAM) laid out the boundaries of the
classical description. KAM theory introduces the theory of chaos into non-integrable
Hamiltonian systems. The pendulum provides the simplest example of the continuous
nonlinear phenomena with which KAM theory is concerned. The breakdown of trajec-
tories heralded by KAM theory is the first proof that the traditional tools of physics
might not be adequate for describing general complex webs.
The KAM theory for conservative dynamical systems describes how the continu-
ous trajectories of particles determined by Hamiltonians break up into a chaotic sea
of randomly disconnected points. Furthermore, the strange attractors of dissipative
dynamical systems have a fractal dimension in phase space. Both these developments
in classical dynamics, KAM theory and strange attractors, emphasize the importance
of non-analytic functions in the description of the evolution of deterministic nonlin-
ear dynamical networks. We briefly discuss such dynamical webs herein, but refer the
reader to a number of excellent topics on the subject, ranging from the mathematically
rigorous, but readable [
35
], to provocative picture topics [
1
] and texts with extensive
applications [
44
].
4.3.1
The logistic equation
The size of the human population grew gradually for about sixteen centuries and then at
an increasing rate through the nineteenth century. Records over the period 1200 to 1700,
while scanty, show some fluctuations up and down, but indicate no significant trend.
