  In this paper an autonomous analytical system of ordinary differential
equations is considered. For an asymptotically stable steady state x0 of the
system a gradual approximation of the domain of attraction DA is presented in
the case when the matrix of the linearized system in x0 is diagonalizable. This
technique is based on the gradual extension of the "embryo" of an analytic
function of several complex variables. The analytic function is the transformed
of a Lyapunov function whose natural domain of analyticity is the DA and which
satisfies a linear non-homogeneous partial differential equation. The equation
permits to establish an "embryo" of the transformed function and a first
approximation of DA. The "embryo" is used for the determination of a new
"embryo" and a new part of the DA. In this way, computing new "embryos" and new
domains, the DA is gradually approximated. Numerical examples are given for
polynomial systems. For systems considered recently in the literature the
results are compared with those obtained with other methods.
