Some remarks on inverse Jacobi multipliers around Hopf singularities

Abstract

We study several properties of inverse Jacobi multipliers V around Hopf singularities of analytic vector fields X in Rn which are relevant to the study of the local bifurcation of periodic orbits. When n=3 and the singularity is a saddle-focus we show that: (i) any two locally smooth and non-flat linearly independent inverse Jacobi multipliers have the same Taylor expansion; (ii) any smooth and non-flat V has associated exactly one smooth center manifold Wc of X such that Wc⊂V−1(0). We also study whether the properties of the vanishing set V−1(0) proved in the 3-dimensional case remain valid when n≥4.