Abstract:

In this talk I will report on a new result on analytic differential systems in $mathbb{R}^n$ with a pair of pure imaginary eigenvalues. We first characterize the existence of either analytic or $mathcal{C}^infty$ inverse Jacobian multipliers of the systems around the origin, which is either a center or a focus on the center manifold. Then we study the cyclicity of the system at the origin through Hopf bifurcation via the vanishing multiplicity of the inverse Jacobian multiplier.