Dulac's problem is a local version of the Hilbert's $16^{th}$ problem. It states that a two dimensional polynomial vector field defines at most a finite number of limit cycles. In fact, it was solved for analytic vector fields by Y. Ilyashenko and J. Ecalle in 1991-1992.

We extend Dulac's problem to real analytic vector fields in $\mathbb{R}^3$.

We say that a real analytic vector field fulfills Dulac's property if

periodic orbits are organized in one of the following configurations:

-There is a neighborhood of the singularity where there are not periodic orbits.

-There is a neighborhood of the singularity in which all periodic orbits are contained in a finite number of invariant surfaces which are filled with periodic orbits.

Both assertions imply local finiteness of limit cycles. In this talk, we prove Dulac's property for three dimensional vector fields with isolated singularity which are perturbations of linear vector fields with a center configuration. That is, for vector fields

$\xi=\left(-by+ A_1(x,y,z) \right)\dfrac{\partial}{\partial x}+\left(bx+ A_2(x,y,z) \right)\dfrac{\partial}{\partial y}+\left(cz+ A_3(x,y,z) \right)\dfrac{\partial}{\partial z}$

where $ b\in \mathbb{R}\setminus \{ 0 \}, \space c\in \mathbb{R}$.