Polynomial normal forms for vector fields on ℝ3

Abstract

The present paper is devoted to studying a class of smoothly (C^∞) finitely determined vector fields on ℝ³. Given any such generic local system of the form $\dot{x}$=Ax+⋯, where A is a 3×3 matrix, we find the minimal possible number i(A) such that the vector field is i(A)-jet determined, and we find the number μ(A) of moduli in the C^∞ classification. We also give a list of the simplest normal forms, that is, polynomials of degree i(A) containing exactly μ(A) parameters.