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 R-3 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 W-C, in particular to know under what conditions W-C subset of V-1(0). To illustrate our results, we solve the center problem for the Lu system. (C) 2012 Elsevier Inc. All rights reserved.