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In-Depth Information
Despite the simplicity of their form, logistic maps provide very rich behavioral
forms. When the parameter
a
is close to a value around 3
57, logistic growths be-
come chaotic. This implies many typical phenomena:
sensitivity
(starting from very
close values, then very different dynamics could be generated),
transitivity
(the dy-
namics takes, in an erratic way, almost all of its possible values), and
dense peri-
odicity
(any value is close, at any distance, to periodical states). A detailed analysis
of chaotic phenomena, and of the chaotic aspect of logistic maps, is outside the
scope of our discussion (see [183]), the literature about logistic map and related as-
pects is very rich and full of spectacular phenomena (very interesting images can
easily found on the web), we refer to [162] for a general analysis of iterated maps
and historical notes. In a very intuitive manner, chaos means impossibility of reli-
able predictions. The surprising fact disclosed by phenomena such as logistic maps
is
deterministic chaos
. The discovery of the existence of this possibility is one of
the deepest mathematical results of the last century, and can be found not only in
complex systems, but in very simple cases, as logistic growth shows. Then, in a de-
terministic chaotic phenomenon, unpredictability is not due to a lack of rule, but to
the impossibility of applying it with the exactness required for a reliable adequacy
to the reality modeled by it.
.
5.6.5
Natural Exponential Growth
Geometrical progressions are growths of exponential type. A pure exponential law
is of kind
a
n
. This happens in a population where, starting from one individual, at
each generation step every individual generates
a
individuals, but no individual at
a generation step can survive at the next generation step. Between arithmetical and
exponential growths there are intermediate levels, for example
power laws
such as
an
2
an
3
. Of course, growths can be modeled with continuous time, and the base
of an exponential law can be lower than one, or equivalently, the exponents can take
negative values. However, here we want to mention the relationship between growth
and the Euler constant
e
, by following an analysis due to the Italian mathematician
of the last century Bruno De Finetti.
Let us consider a simple process where in a unitary time an initial quantity
a
0
be-
comes 2
a
0
. We can assume that this process is due to a uniform increment along the
considered time interval. This means that if we partition the interval into
n
subin-
tervals, then during each of them a quantity
a
0
/
,
,...
n
is added to the initial quantity. In
fact,
a
0
+
2
a
0
. Now, let us change this increment rule, by assuming that
at each step the increment by the 1
n
(
a
0
/
n
)=
n
factor is not applied to the initial quantity,
but to the global quantity cumulated along the preceding steps. We can express this
natural growth rule
by the following inductive definition (
i
/
>
0):
a
i
+
1
=
a
i
+
a
i
/
n
this means that:
a
i
+
1
=
a
i
(
1
+
1
/
n
)
