Applied Maths Seminar: Dullin -- Normal forms for volume preserving maps and bifurcations of invariant circles

Volume preserving maps are related to divergence free vector fields (and sl(n)) in the
same way as symplectic maps are related to Hamiltonian vector fields (and sp(2m)).  I
will give an introduction to the dynamics of volume preserving maps (with n=3)
highlighting similarities and differences to the symplectic case.  In particular I will
show that for maps with nilpotent linearisation there exists an optimal normal form in
which the truncation of the normal form expansion at any order gives a map that is
exactly volume preserving with an inverse that is also polynomial and has the same
degree (Physica D 237:156).  Using the unfolding of this normal form we study the
Saddle-Node-Hopf bifurcation (one eigenvalue 1, two complex conjugate eigenvalues on the
unit circle) in which an invariant circle is created.  A numerical study of the normal
and transverse frequencies of these invariant circles under parameter variation reveals
new types of bifurcation when resonances are encountered.  Some of the invariant sets in
the dynamics show a striking similarly to structures found in vortex rings and the
collision of vortex rings in fluids.