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Colloquium "On stability and instability is Hamiltonian systems"


Tid: 2018-06-13 15:30 till: 16:30
Plats: MH:Hörmandersalen, Matematikcentrum
Kontakt: mikael [dot] persson_sundqvist [at] math [dot] lth [dot] se
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Speaker: Maria Saprykina, KTH

Here is a heuristic description of the circle of problems we are interested in. Imagine a chain of mathematical pendula attached to the wall in a line, and moving. If they are not coupled, the energy of each pendulum is preserved for all time. Now we join each pair of neighboring pendula by a very thin rubber band. Of course, the total energy of the system is still preserved.
But what happens with the energy of each individual pendulum? KAM theorem asserts that under some generic assumptions, for "most'' initial conditions the energy of each pendulum will stay close to the initial one for all time. This phenomenon is called KAM-stability. We will mention some recent results describing the necessary conditions for KAM-stability as well as their sharpness.
On the other hand, what happens for the "small part'' of the initial conditions that are not descrided by this theorem? One of our results states that there exist initial conditions and a sequence of moments of time, such that at j-th moment of time the j-th pendulum moves with almost the total energy of the system. This behaviour is a manifestation of so-called Arnold diffusion. I shall speak about one more example exhibiting Arnold diffusion.
These two results were oblained in collaboration with Vadim Kaloshin and Mark Levi.