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The qualitative theory of differential equations is closely linked with the
study of vector fields. Already on the sphere / plane, one can see examples of
non-trivial behavior of solutions (phase curves of a vector field). Moreover, the
complexity of degeneracy ranges from simple cases that will be discussed in the
course, to open mathematical problems In the light of possible physical
applications and computer simulations, it is natural to consider not only the
behavior of individual orbits, but also the behavior of typical points in the
Lebesgue measure («random solutions»), and, accordingly, the attractors and
invariant measures that emerges in this case. The course is dedicated to this
area.
- Vector fields on the plane/sphere. Omega-limit sets, singular points, limit
cycles. Morse-Smale systems. Poincaré-Bendixson theorem.
- The Andronov-Pontryagin criterion and the structure of a typical vector field.
- The concept of codimension of degeneration. Sotomayor's theorem, examples
of degeneration.
- Normal forms of saddle and saddle-node.
- Attractors in measure theory: Milnor attractor, statistical and minimal
attractors. SRB measure. The Krylov-Bogolyubov theorem.
- Codimension two: biangle. Bowen's example. The absence of time (Birkhoff)
averages for almost all points with respect to the Lebesgue measure.
- Bifurcation diagrams. Bifurcation diagram «biangle».
- Codimension three: modified Bowen's example. Kleptsyn's theorem on the
non-coincidence between the statistical and minimal attractors.
- Direct products of flows on a sphere: the Agraval-Rodriguez-Field theorem on
the behavior of a measure near a product of homoclinic loops.
- Possible singular points and polycycles of codimension two. A complete
description of the statistical attractor for the product of flows of codimension
two. The product of flows on a sphere of codimension three: the appearance of
synchronization in the absence of interaction.
 Exam
 RSS: Forthcoming seminars
Lecturers
Dukov Andrei Valerevich
Minkov Stanislav Sergeyevich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
