Normal forms for Volume Preserving Maps:
Everything you wanted to know about this year's Applied Math T-shirt
James Meiss
Applied Mathematics
Volume-preserving maps arise in fluid dynamics and are also a natural generalization of area-preserving maps. Understanding their dynamics is crucial for developing a theory of three-dimensional fluid mixing. Holger Dullin and I have been studying the normal forms for such maps near special fixed points, namely a fixed point with two eigenvalues on the unit circle and a third which is one, or a fixed point with three unit eigenvalues. The unfolding of these normal forms give rise to bifurcations that create invariant tori, and some of these can be seen on the APPM t-shirt this year. I will explain some of these structures and some of the open questions that we are trying to solve.
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