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 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 with the same degree.

Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

“…The three-dimensional analogue of the two-dimensional, area-preserving Hénon map can be found by normal form expansion and unfolding near a triple-one multiplier [DM08]; we believe this quadratic map, (1), should serve as the prototype for volume-preserving dynamics in R 3 .…”

Bifurcations in Volume-Preserving Systems

“…From §2.1 we know that a triple eigenvalue 0 with a single Jordan chain has codimension 2 as well. The nonlinear unfoldinġ [18] not only contains the volume-preserving Hopf bifurcation in a subordinate way, but also a normally hyperbolic saddle-node bifurcation, compare with [19].…”

Symmetry reduction by lifting for maps

“…(34) Similar maps on R 3 , arise as the normal form for a volume-preserving map near saddle-node bifurcation with a triple-one multiplier [DM08]. The map (34) has symmetry Y = ∂ ∂x which generates the flow ψ t (x, y, z) = (x+t, y, z).…”