Tudor Ratiu, Ecole polytechnique fédérale de Lausanne
This talk will present generalizations of the Weinstein-Moser theorem about the existence of periodic orbits around stable equilibria of Hamiltonian systems. The results discussed deal with symmetric Hamiltonian systems and address the existence and the lower estimates on the number of bifurcating relative equlibria and relative periodic orbits emanating from equlibria and relative equlibria. The case of formally unstable critical elements will also be discussed. Techniques of singular reduction are used to obtain some of these results. Special emphasis will be put on the so called ``reconstruction equations'' which mirror the equations of motion in the singularly reduced spaces but expressed on the original smooth manifold. The results are obtained by combining techniques of singular reduction, normal forms, and topological estimates. If time permits, the symmetric Hamiltonian Hopf bifurcation will be shown to be obtainable by similar methods which brings it closer in formulation to the classical dissipative Hopf bifurcation theorem.