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Inverse Jacobian multipliers and Hopf bifurcation on center manifolds

In this paper we consider a class of higher dimensional differential systems in $\mathbb R^n$ which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of either analytic or $C^\infty$ inverse Jacobian multipliers of the systems around 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 Hopf bifurcation by using the vanishing multiplicity of the inverse Jacobian multiplier.