Examples of Hamiltonian systems with Arnold diffusion

Homogenization and Random Phenomenon

27 November 14:00 - 14:55

Maria Saprykina - KTH Royal Institute of Technology

Here is a heuristic description of the problem setting in ``physical terms''. 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 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. But 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 results were oblained in collaboration with Vadim Kaloshin and Mark Levi.
Henrik Shahgholian
KTH Royal Institute of Technology
Panagiotis Souganidis
The University of Chicago


Henrik Shahgholian

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