Nel periodo dal 7 al 17 maggio sarà ospite del Centro De Giorgi la
Prof.ssa Lai-Sang Young (Courant Institute of Mathematical
Sciences, New York University). In questo periodo terrà un corso dal titolo "A
mathematical theory of strange attractors" in 5 lezioni. Il corso è
principalmente rivolto, per il livello, agli studenti delle scuole di dottorato.
Seguono i
dettagli:
Orario: 9 maggio ore
16-18
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Descrizione:
In this course, I would like to present some parts of a mathematical theory of
rank one attractors. By "rank-one attractors", I refer to attractors that have
one unstable direction (hence they are chaotic) and strong contraction in all
other directions. Hénon attractors (with small b) are prototypical examples, but
the class I will discuss is considerably more general: They can live in phase
spaces of any dimension $\geq 2$ and have been shown to appear in natural
contexts arising from mechanics and phyiscs.
A tentative outline of the
lectures is as follows
Lecture 1. Relevant 1D dynamics
Lecture 2.
Geometric and structural differences between 1D maps and a class of rank-one
attractors Because of the strong contraction, rank-one attractors have a one
dimensional character. This allows us to study them by leveraging our knowledge
of 1D maps.
In my first lecture, I will review those aspects of 1D dynamics
that are relevant, and in my second lecture, I will stress the differences
between 1D systems and systems defined by higher dimensional maps that are small
perturbations of 1D maps. Both similarities and differences are striking. To
give an example of the latter, in higher dimensions the counterpart of critical
points in 1D are fractal sets.
Lectures 3 and 4. SRB measures
I will
begin with an introduction to SRB measures, what they are, why they are
important, and why Axiom A attractors admit them. This leads to the question of
existence of
SRB measures in the nonuniform hyperbolic category, a
question that is poorly understood. In fact, the attractors introduced in
Lecture 2 are among the few situations for which
SRB measures have been
shown to exist. In Lecture 4, I will give some idea of what this proof entails.
Material from Lectures 1,2 and 3 will be used.
Lecture 5.
Applications
We have identified a class of attractors and discussed their
statistical properties, but so far have not touched upon the question of their
existence(!) let alone where these attractors can be found. In this final
lecture, I will state (without proof) a theorem that guarantees the presence of
these attractors when a concrete set of conditions are met, and demonstrate how
to verify these conditions in some scenarios encountered frequently, such as the
periodic forcing of limit cycles or systems undergoing Hopf
bifurcations.