What is new about chaotic attractors

Supervisor
Maria José Pacifico - Instituto de Matemàtica, UFRJ (Universidade Federal do Rio de Janeiro)
Date and time
Monday, May 28, 2012 at 2:30 PM - Aula didattica E (Piano terra, Ca' Vignal 1)
Programme Director
Gaetano Zampieri
External reference
Publication date
May 21, 2012
Department
Computer Science  

Summary

I will survey on flows presenting equilibria attached to regular orbits. 
The classical example for this is the Lorenz-attractor flow as well
the geometric Lorenz flow.
More than three decades passed before the existence of the Lorenz
attractor was rigorously established by Warwick Tucker with a
computer-assisted proof in the year 2000.

The difficulty in treating this kind of systems is both conceptual and
numerical. On the one hand, the presence of the singularity accumulated by
regular orbits prevents this invariant set to be uniformly hyperbolic. On
the other hand, solutions slow down as they pass near the saddle
equilibria and so numerical integration errors accumulate without bound.
Trying to address this problem, a successful approach was developed by
Afraimovich-Bykov-Shil'nikov and Guckenheimer-Williams independently,
leading to the construction of a geometrical model displaying the main
features of the behavior of the solutions of the Lorenz system of
equations.

In the 1990‚s a breakthrough was obtained by Carlos Morales, Enrique
Pujals and Maria José Pacifico following very original ideas developed
by Ricardo Mañé  during the proof of the C1-stability conjecture,
providing a characterization of robustly transitive attractors for
three-dimensional flows, of which the Lorenz attractor is an example.





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