Integrability and chaotic
attractors.
We have introduced a method to study the chaotic attractors of
different systems by using the integrability informations existing
for these systems. This method provides us with algebrical
bounds for the extension of chaotic attractors in phase space
(which is interesting because in case of chaotic motion, we do not
know the expression of the chaotic trajectory). The method also
yields ranges of parameter values for
which there is no chaotic motion.
As for future work, it would be interesting to know if there is
a underlying link between the integrability
notions in a system and its chaotic behaviour via the
semi-permeables surfaces (see
here, here,
and here). Such links already exist in dynamical
systems, as for example the relations between real time or complex
time singularities and turbulence have alrerady been noticed in
the Euler or Navier Stokes equations.
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