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internal interactions of the particles inside an atom, in a chemical process, in a cy-
clone, in an economic process of resource exchange, in the course of an epidemic,
or in the development of an organism.
The Greek philosopher Eracyitus says, “[...] Panta rei” (
`
`
), that is,
“everything is changing” or also “existence is change”. But if change follows a rule,
something has not to change. In fact, in order to localize and describe a process
as an entity, something has to remain stable during it. Existence is a mysterious
mixing of variation and invariance underlying objects and events, at each level of
reality. A
dynamical system
is a structure hosting a dynamics. The term dynamics
points to the process while the term system points to the structure. A river, a city,
a living organism, are always different in time, nevertheless, their individualities
persist unchanged during their lives.
Since the epochal discovery of the laws of planetary motions, dynamical systems
have been mathematically studied by means of differential equations. Recently the
study of complex systems arising from life sciences, economics, meteorology, and
many other fields, requires novel ideas and different approaches for the analysis and
modeling of a wider class of dynamical systems, and for a great variety of aspects
concerning the dichotomy structure/behavior. In particular, discrete dynamical sys-
tems seem to open new modeling possibilities and pose new problems where the
algorithmic aspects replace the geometrical and differential perspective of classical
dynamics.
In 1684 the astronomer Edmund Halley traveled to Cambridge to ask Newton
about planetary motions. It was known that planets move around the Sun in ellipses.
The question Halley posed to Newton was about the reason for this elliptic shape. Sir
Isaac replied immediately: “I have calculated it”. The general rigorous proof of this
phenomenon was definitively given by Euler in 1749 (elliptic shape is a consequence
of the inverse-square law and of the fundamental law of dynamics) [165].
Halley's question is a first example of the
dynamical inverse problem
,thatis,
given an observed behavior, is there a mathematical system that can explain this
behavior? This is a crucial issue, in fact, when we can answer this question, then we
“dominate” the system because we can predict or alter its behavior.
Any system changing in time can be identified with some variables. As they de-
fine a system, they must verify some relations which are the
invariant
of the behav-
ior. Newton's solution to Halley's question was obtained first by determining some
invariants of the planetary orbits, specified in terms of differential equations, then,
by solving these equations, the mathematical form of these orbits was deduced. In
general, in order to define a mathematical system related to a process, we need to de-
termine its
variables
and
invariants
. This schema of analysis for dynamical systems
is very general and continues to hold in different formal frameworks of dynamical
representations. To suggest the wide range of this paradigm, let us consider the situ-
ation of a discrete phenomenon. Suppose we observe a device generating as output
words (in a particular alphabet). A very natural question is: which is the grammar
of the language generated by this device? And, in which manner does its internal
structure realize the generation of these words? If we assume that any event can
be discretely represented by a suitable word, then the search for rules underlying
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