Multiple Hopf bifurcation in R3R3 and inverse Jacobi multipliers

In this paper we study the maximum number of limit cycles that can bifurcate from a singular point of saddle-focus type of an analytic, autonomous differential system in R3R3 under any analytic perturbation that keeps the location and nature of the singularity. We only consider those foci on center manifolds having associated two nonzero purely imaginary and one nonzero real eigenvalues. Our approach is different from the classical one in the sense that we do not use any center manifold reduction to compute Poincaré–Lyapunov constants. Instead, we study the multiple Hopf bifurcation first doing a Lyapunov–Schmidt reduction to the associated Poincaré map, obtaining in this way an analytic reduced displacement map. Next we prove that the order of this displacement map coincides with the vanishing multiplicity (denoted m) of any locally smooth and non-flat inverse Jacobi multiplier. Finally the cyclicity of the focus is given in terms of m.