course: A mathematical theory of strange attractors
speaker: Lai-Sang Young (New York University)abstract: 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.
timetable:
Tue 9 May, 16:00 - 18:00, Aula Dini
Wed 10 May, 16:00 - 18:00, Aula Dini
Thu 11 May, 15:00 - 17:00, Aula Dini
Mon 15 May, 16:30 - 18:00, Aula Dini
Tue 16 May, 16:30 - 18:00, Aula Dini
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