Some remarks on inverse Jacobi multipliers around Hopf singularities

We study several properties of inverse Jacobi multipliers V around Hopf singularities of analytic vector fields X in R n 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 W c of X such that W c ⊂ 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 .