Biomedical Engineering Reference
In-Depth Information
scale),
s
2
/
s
1
, and not directly on the absolute scale of observations,
s
1
or
s
2
ps
2
/
ps
1
=
(
s
1
)
ε
q
2
/
q
1
=
s
2
/
(2.4)
s
1
is a discrete variable, while for statisti-
cal fractals, like those in nature,
s
2
/
For exact fractals,
s
2
/
s
1
may change in a continuous
fashion while not violating the validity of the equation (2.4). For
statistical fractals, the two sides of the equation are equal only in
distribution (note that
=
d
should be used instead of
=
here)
q
1
=
d
ps
2
/
ps
1
=
d
(
s
1
)
ε
q
2
/
s
2
/
(2.5)
For natural fractals, scale invariance holds only for a restricted
range of the absolute scale
(10)
. The upper limit of validity
s
max
,
for natural fractals falls into the range of the size of the structure
itself, likewise the lower limit,
s
min
, falls into the dimensions of the
smallest structural elements (i.e. there are no more newer details
when the resolution is increased further). The scaling range, SR,
is then given in decibels.
SR
=
log
10
(
s
max
/
s
min
).
(2.6)
3. Complex
Systems
A complex system has a great deal of structural and/or functional
redundancies and which, in turn, lend it a robust behavior, where
quite a few of its elements may be dropped without compromis-
ing the structural or functional integrity of the complex system.
In the absence of such robust behavior, the system can still be
regarded as “complicated.” In this system, however, excluding
one key element can bring the whole system to a halt.
Another key behavior of a complex system is that it can adapt
to changing conditions often in producing a non-linear response,
a very hard to predict behavior. The description of a complex
system's structure and function requires special tools such as
those used in chaos and/or fractal theories
(8)
. Chaos theory
describes the non-linear dynamics emerging from coupled dif-
ferential equations and predicts the behavior of the system in
Newtonian space which is characterized by a fractal attractor
(4)
.
The fractal approach is aimed at identifying the presence of self-
similarity in the design of complex structures or in the dynam-
ics of complex temporal processes, where both manifestations
are correlation of patterns
(11)
. Complex systems in nature can
be viewed as implementations of nature's “blueprints” employ-
ing fractal spatiotemporal dynamics, keeping in mind that frac-
tal dynamics is only a model which can indeed describe many
