Μάθημα : ΣΥΝΗΘΕΙΣ ΔΙΑΦΟΡΙΚΕΣ ΕΞΙΣΩΣΕΙΣ ΚΑΙ ΔΥΝΑΜΙΚΑ ΣΥΣΤΗΜΑΤΑ , Χειμερινο 2024-2025

Την ερχομενη Παρασκευη εχουμε μια ομιλια επανω στη χαωτικη συμπεριφορα σε δυναμικα συστηματα, στις 3 διαστασεις. Το θεμα σχετιζεται με τα οσα μας ειπε ο Δημητρης ο Εμμανουηλ.

ομιλητης αυτην την Παρασκευη ειναι ο  Eran Igra, που πηρε το διδακτορικο
του απο το πολυτεχνειο της Haifa, και ειναι επισκεπτης στο the Shanghai
Institute for Mathematics and Interdisciplinary Sciences, China.
Το θεμα του ειναι Δυναμικα Συστηματα.

Τιτλος: Essential dynamics in chaotic attractors

Συνοψη: Assume we have a three-dimensional flow whose dynamics are
hyperbolic on some attracting invariant set. Whenever this is the case,
the dynamics on the attractor will be chaotic – for example, the attractor
will include infinitely many periodic orbits, the dynamics will be stable
under perturbation, along with many other interesting qualitative and
quantitative properties. As hyperbolic dynamics can be thought of as an
idealized version of chaotic behavior and since chaotic attractors appear
almost whenever nonlinear dynamics are involved, one is led to the
following question – just how much of this rich and beautiful theory of
hyperbolic dynamics can be applied to study real-life chaotic attractors,
i.e., chaotic attractors for flows which were discovered through science
and engineering? Currently, the answer is “not much”. To date, only one
“real life” chaotic attractor was proven to generate hyperbolic dynamics,
and in many other interesting models we have good indications the dynamics
will probably not be hyperbolic.
And yet, per most numerical simulations, real life chaotic attractors
often behave as if they were, in fact, hyperbolic – which led to the
famous “Chaotic Hypothesis”, originally due to Giovanni Gallavotti: the
dynamics on chaotic attractors are essentially hyperbolic. Inspired by the
Thurston-Nielsen Classification Theorem and the Orbit Index Theory, we
give a partial answer to the Chaotic Hypothesis. Namely, we prove that
whenever certain heteroclinic conditions are satisfied by a given flow,
the topology of the resulting phase space will force chaotic behavior –
which can be thought of as “essentially hyperbolic behavior”. Following
that, we prove how these results can be applied to study two famous
examples of chaotic attractors – the Lorenz and the Rössler attractors.
Time permitting, we will conjecture how these results can possibly be
generalized to derive a more general topological forcing theory for
three-dimensional flows.