Speaker: Leonid Bunimovich
Title: Mushrooms and Generic Behavior of Hamiltonian Systems
Abstract: Among the greatest achievements in dynamics in the last centure there were the proofs of stability (under small perturbations) both of chaotic and of a regular (integrable) behavior of dynamical systems (Kolmogorov-Arnold-Moser theory). Generic Hamiltonian systems though are neither chaotic nor integrable. Instead they demonstrate a mixed behavior when the islands of stability (integrability) in their phase (coordinate) space are situated in a chaotic sea. This picture of divided phase space has been obtained in many numerical and real experiments but never has been proven to exist in some natural and visible examples. The main difficulty is to exactly know how a boundary between chaotic and regular regions look like. I'll describe the first very simple examples where a picture of coexistense can be proven absolutely elementary. This new class of systems is called mushrooms. A mushroom is a generalization of a stadium. As a byproduct it is shown that islands of stability do not need to form a (multy-) fractal (a usual assumption in various theories. Another byproduct is an elementary construction of systems where any (finite or infinite) number of islands coexists with any (finite or infinite) number of chaotic seas. Graduate students are mostly welcomed; a talk will be understandable to everybody familiar with recktangulars, circles and ellipses.