Type: Seminar

Distribution: World

Expiry: 23 Apr 2008

Auth: wm@p6283.pc.maths.usyd.edu.au

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.