Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

Abstract

We introduce a sl 2 -invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the family constitutes a Lie algebra structure and each vector field from this family is solenoidal, completely integrable and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie algebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this family can be uniquely characterized by two alternative representations; one uses a vector potential while the other uses two functionally independent Clebsch potentials. Our normal form results are designed to preserve these structures and representations. The results are implemented in Maple in order to compute vector potential and the Clebsch potential normal forms of a given vector field from this family. Some practical normal form coefficient formulas for degrees of up to four are presented.

Versal normal form for nonsemisimple singularities

“…Note that due to the assumptionm 2,1 ,m 3,2 ∈ R * the transformation given by (4.2) when ε = 0 is invertible. Now, we writing down (4.3) in terms of vector fields from A , B, C given in [8] to find the following…”

On the representations andZ2-equivariant normal form for solenoidal Hopf-zero singularities

“…In [19] the authors express a Lie algebra of completely integrable solenoidal triple-zero singularities via Euler's form and vector potential. As in their studies, due to the solenoidal property of L , here we shall present any vector field in this Lie algebra using vector potentials and Euler's form.…”

“…In [2] the Z 2 -equivariant normal form for Hopf-zero vector fields are computed under the assumption that the cubic terms be non-zero, for results on the bifurcation of Hopf-pitchfork singularities, see [3][4][5]. More studies regarding the normal form of dynamical system could be found in [14,15,19,26].…”

Equivariant decomposition of polynomial vector fields

“…The complexities of this programme are already clear in the treatment of the 3D irreducible nilpotent singularity in [6]. There the structure constants are computed using Maple computations and extrapolation, a time consuming method which does not seem to be well suited for higher dimensions.…”