Research in Dynamical Systems

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Professor Franks will continue his investigation of the dynamics of surface maps, especially area preserving ones. Questions concerning the existence of periodic behavior in area preserving two-dimensional discrete dynamical systems will be addressed. Professor Xia's proposed research concentrates on the following three aspects of dynamical systems: (1) the dynamics of the Newtonian n-body problem and celestial mechanics; (2) hyperbolicity and bifurcations in global dynamics; and (3) Hamiltonian dynamics and symplectic diffeomorphisms. Surface maps are two-dimensional transformations; the study of such transformations, especially area preserving ones, has a long history going back to Poincare and G. D. Birkhoff. There are numerous applications of results in this area to classical mechanics as well as to more modern chaotic dynamics. In physical terms, Professor Xia's proposed research is on: the dynamics of bodies being acted on by gravitation; the way the dynamics of a system changes as parameters in it vary; and the dynamics of systems in which energy is conserved.