Existence of inverse Jacobi multipliers around Hopf points in R3: Emphasis on the center problem

We consider the center problem at Hopf points of analytic systems in R3 that has a classical solution in the Lyapunov Center Theorem which is given in terms of an analytic first integral. Here we give a new solution in terms of an analytic inverse Jacobi multiplier V. The existence of a smooth and non-flat inverse Jacobi multiplier around a Hopf point of saddle-focus type is also proved. When studying these problems, we needed to discuss the relation between inverse Jacobi multipliers and center manifolds Wc, in particular to know under what conditions Wc⊂V−1(0). To illustrate our results, we solve the center problem for the Lü system.