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

TL;DR

This paper develops a simplified normal form for divergence-free vector fields and volume-preserving maps near equilibrium points with nilpotent linearization, facilitating analysis of such systems.

Contribution

It introduces a novel normal form for divergence-free vector fields and volume-preserving maps with nilpotent linearization, enabling polynomial inverse and volume preservation at any truncation degree.

Findings

Normal form expressed as a single function of two variables in the third component.

Truncated normal forms are exactly volume-preserving with polynomial inverses.

Applicable to systems with maximal Jordan block nilpotent linearization.

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.