I have a system of odes $\dot{\mathbf{x}} = \mathbf{f(x)}$ where $\mathbf{x} \in \mathbb{R}^{2}$ and $\mathbf{f(x)}$ is defined below:
$$\dot{x} = x- y - x^{3}, \qquad \dot{y} = x+y-y^{3}$$
I would like to apply Dulac's criteria to show that there is a limit cycle in the annulus $1 < || \mathbf{x}|| < \sqrt{2}$. Thus, I try to find a function $B(x,y) \mapsto \mathbb{R}$ such that $\nabla \cdot (B \mathbf{f})$ is not identically zero or change signs in a region $E = \{\mathbf{x}: 0 \leq ||\mathbf{x}||< \sqrt{2} \}$.
How could I obtain a suitable test function $B$ for this?