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.

References in corpus (6)

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• Box products in nilpotent normal form theory: The factoring method
• Analytic normalization of analytically integrable differential systems near a periodic orbit
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Cited by in corpus (2)

• On the representations and ${\mathbb{Z}}_2$-equivariant normal form for solenoidal Hopf-zero singularities
• Versal Normal Form for Nonsemisimple Singularities