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5
Non-analytic dynamics
In this chapter we investigate one procedure for describing the dynamics of complex
webs when the differential equations of ordinary dynamics are no longer adequate, that
is, the webs are fractal. We described some of the essential features of fractal func-
tions earlier, starting from the simple dynamical processes described by functions that
are fractal, such as the Weierstrass function, which are continuous everywhere but are
nowhere differentiable. This idea of non-differentiability suggests introducing elemen-
tary definitions of fractional integrals and fractional derivatives starting from the limits
of appropriately defined sums. The relation between fractal functions and the fractional
calculus is a deep one. For example, the fractional derivative of a regular function yields
a fractal function of dimension determined by the order of the fractional derivative.
Thus, changes in time of phenomena that are described by fractal functions are prob-
ably best described by fractional equations of motion. In any event, this perspective is
the one we developed elsewhere [
31
] and we find it useful here for discussing some
properties of complex webs.
The separation of time scales in complex physical phenomena allows smoothing over
the microscopic fluctuations and the construction of differentiable representations of
the dynamics on large space scales and long time scales. However, such smoothing is
not always possible. Examples of physical phenomena that resist this approach include
turbulent fluid flow [
23
]; the stress relaxation of viscoelastic materials such as plastics,
rubber and taffy [
21
]; and finally phase transitions [
17
,
27
] as well as the discontinuous
physical phenomena discussed in the first chapter. Recall that the physical phenomena
included the sizes of earthquakes, the number of solar flares and sunspots in a given
period, and the magnitude of volcanic eruptions, but also intermittent biological phe-
nomena, including the variability of blood flow to the brain, yearly births to teenagers
and the variability of step duration during walking.
Metaphorically, complex phenomena, whose dynamics cannot be described by ordi-
nary differential equations of motion, leap and jump in unexpected ways to obtain food,
unpredictably twist and turn to avoid capture, and suddenly change strategy to anticipate
changes in an enemy that can learn. To understand these and other analogous processes
in the social and life sciences, we find that we must adopt a new type of modeling, one
that is
not
in terms of ordinary or partial differential equations of motion. It is clear
that the fundamental elements of complex physical phenomena, such as phase transi-
tions, the deformation of plastics and the stress relaxation of polymers, satisfy Newton's
laws. In these phenomena the evolution of individual particles is described by ordinary
