Applied Maths Seminar: Dullin -- Vanishing twist near resonant Hamiltonian equilibria

Holger Dullin, School of Mathematics and Statistics, University of Sydney 

I will review the relation between bifurcations of periodic orbits of area preserving
maps and the vanishing of twist, and then describe new results that establish a similar
connection for resonant equilibria, in particular the 1:-1 (Hamiltonian Hopf) and 1:-2
resonances.  The main result is that the rotation number has a critical value when the
quadratic part of the energy is positive and energy in higher order terms is vanishing.
This once again shows that the vanishing of twist is generic in one-parameter families
of Hamiltonian systems with two degrees of freedom.