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In this paper we consider a class of higher dimensional differential systemsin $\mathbb R^n$ which have a two dimensional center manifold at the originwith a pair of pure imaginary eigenvalues. First we characterize the existenceof either analytic or $C^\infty$ inverse Jacobian multipliers of the systemsaround the origin, which is either a center or a focus on the center manifold.Later we study the cyclicity of the system at the origin through Hopfbifurcation by using the vanishing multiplicity of the inverse Jacobianmultiplier.

Inverse Jacobian multipliers and Hopf bifurcation on center manifolds