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Holger Dullin
School of Mathematics & Statistics, University of Sydney
Normal forms for volume preserving maps and bifurcations of invariant
circles
Wednesday 23rd April 14:05-14:55pm,
Eastern Avenue Lecture Theatre.
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.
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