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

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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References in corpus (2)

• Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation
• Heteroclinic intersections between Invariant Circles of Volume-Preserving Maps

Cited by in corpus (6)

• Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations
• Accelerator modes and anomalous diffusion in 3D volume-preserving maps
• Visualizing Attractors of the Three-Dimensional Generalized Hénon Map
• Anti-Integrability for 3-Dimensional Quadratic Maps
• Symmetry Reduction by Lifting for Maps
• Dynamical systems on the Liouville plane and the related strictly contact systems