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Non Linear Dynamics and Chaos

A system is an ensemble of components well delimited in space and
time; outside the system is the external world. lf thr-rc is no mass
transfer across the delimiting surface the system is said to be
~closed“: if tbrrc is no heat transfer ~ or more generally no radiation
escnanged with the external world. the system is said to be "isolated".
The complexity of the system is difficult to precise: it imp..:s the
specification of observability scales: a trivial etample is the
difference in the concept used for the engineer who designs a turbine
and the physicist who studies molecular transformations; both of them
use the entropy concept. For the engineer. entropy S is defined as
dS=dOIT 0 heat flux exchanged T absolute temperature: for the
physicist entropy is the logarithm of the probability that an event of
a given complexiont in terms of molecule arrangement) happens. This
notion of scale is of prime importance as we will see later.
The conelaticn concept is Very useful in the study of systems t see.
for example. the proceedings of the International Symposium on "The
correlation" rel'l2lt.
Correlation apparently bears different meanings for mathematicians.
physicists and engineers. The correlation is commonly‘used as a
device to quantify an uncertainty in physics phenomena. The
uncertainty can be a fundamental one. like in rh._ quantum mechanics.
or can he produced by the large number of parameters which are
considered. or it can simply be an appearance like in phenomena
linked to the determinist chaos. To understand such situations. it is
nc:cssary to enlarge the concept of correlation and even to go beyond
its limits (concept of linkage. resemblance. distance). That's why it
seems worthwhile considering the question.
Finally the perception and comprehension mechanism of the human
brain. as well as the. "neuronal maehincs" which are being developed.
For further details see ref Ill the paper presented by A. Favre
"Correlations Spatio-Ternporelles. determinism: et chaos".
encourage us to eunrine the importance ot thc part played by the
correlation.
The next concept to be intrraluccd is order and disorder. Complete
disorder is imlepcndcncc. All other definitions are merely negative
such as this one: a sufficient (but not necessary) condition to say that
a system is not disorganiled is that correlations between the
fluctuating parameters - if any are not null.