Nilpotent normal form for divergence-free vector fields and volume-preserving maps

H. R. Dullin; J. D. Meiss

doi:10.1016/j.physd.2007.08.014

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arXiv · 0706.1575

Nilpotent normal form for divergence-free vector fields and volume-preserving maps

H. R. Dullin·J. D. Meiss

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Abstract

We study the normal forms for incompressible flows and maps in the neighborhood of an equilibrium or fixed point with a triple eigenvalue. We prove that when a divergence free vector field in $\mathbb{R}^3$ has nilpotent linearization with maximal Jordan block then, to arbitrary degree, coordinates can be chosen so that the nonlinear terms occur as a single function of two variables in the third component. The analogue for volume-preserving diffeomorphisms gives an optimal normal form in which the truncation of the normal form at any degree gives an exactly volume-preserving map whose inverse is also polynomial inverse with the same degree.

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BibTeX RIS

H. R. Dullin, J. D. Meiss. 2007-06-11. Nilpotent normal form for divergence-free vector fields and volume-preserving maps. https://doi.org/10.1016/j.physd.2007.08.014

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