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Лекция академика А.Т.Фоменко и В.А.Кибкало на Летней математической школе ведущих университетов Азии: “Topological approach to integrable Hamiltonian systems and its bifurcation analysis“ Abstract For a wide class of dynamical systems, phase space of such a system can be equipped with a symplectic structure or Poisson bracket. In this case, a dynamical system can be defined by a Hamiltonian vector field. The integrability of such a system means that the Hamiltonian vector field has a “sufficient” number of independent first integrals. Thus the phase space M^2n can be stratified (foliated) into common level surfaces of these first integrals. According to the Liouville theorem, typical fiber is a Lagrangian n-dimensional torus and each phase trajectory on it is dense. Singular points and trajectories of the system belong to “singular“ common levels, in the neighbourhood of which the foliation is not trivial, i.e. contains a bifurcation. In our lecture, we will discuss several results about such bifurcations(singularities) and their impact on the qualitative properties of dynamic systems. Particularly, we will discuss an approach of topological graph invariants invented and developed by for integrable Hamiltonian systems with 2 degrees of freedom and its applications to systems of rigid body dynamics and billiards on domains bounded by confocal quadrics.
