I have the following system
\begin{equation} \begin{aligned} \dot{x} & {}={} 1 + x^2y - 3x = f(x,y) \\ \dot{y} & {}={}2x - yx^2 = g(x,y) \end{aligned} \tag{1} \end{equation}
with one equilibrium point $A(1,2)$. After linearizing around $A$, I obtain the system
\begin{equation} \begin{aligned} \dot{x} & = x + y \\ \dot{y} & = -2x - y \end{aligned} \tag{2} \end{equation}
The eigenvalues of the matrix $J$ of system (2) are $λ_{1,2} = \pm \mathrm{i}$. Because the origin is center for system (2) we can't apply Hartman-Grobamn Theorem. But can I do this instead?
Let us consider $F=(f,g)$. Then $\mathrm{div}F = 2xy - 3 - x^2 < 0$ around $A(1,2)$ if we consider a region $S = \{ (x,y) \in \mathbb{R}^2: \frac{1}{2} \leq x \leq \frac{3}{2} \ \mathrm{and} \ \frac{3}{2} \leq y \leq \frac{5}{2} \}$. Then, the equilibrium point $A$ will be an attractor. Thus, $A$ is asympotically stable focus. Is this right or wrong? Do we know that if the origin is center for the linearized system then the equilibrium point for the nonlinear system will be either a center or a focus?
Thanks for any help / guidance.