Unique periodic orbit for ODE consequence of Poincaré–Bendixson and Dulac’s Criterion The Chicone exercise 1.201 asks prove that the system
$$\dot x=x+y-x^3=g(x,y)$$ $$\dot y=x-y-y^3=h(x,y)$$
has a unique globally attracting limit cycle on the punctured plane.
I prove that if exist a function $B(x,y)$ such that $(Bg)_x+(Bh)_y$ is not identically zero and of fixed sign on annular set then exist a unique periodic orbit. 
This result should make things easier but I still can not see what is the function I need. If you have any idea or suggestion I would help

First you can use Poincare-Bendixon and Dulac’s Criterion to obtain a limit cycle and then use the following argument the limit cycle is a global attractor. Let $u=x^2+y^2$. Then
\begin{eqnarray}
u'&=&2xx'+yy'\\
&=&2[x^2+2xy-y^2-(x^4+y^4)]\\
&\le&2[x^2+2xy+y^2-\frac{(x^2+y^2)^2}{2}]\\
&=&2u-u^2\\
&=&u(2-u).
\end{eqnarray}
For $u\in(0,2)$, we have
$$ \frac{u'}{u(2-u)}\le 1.$$
Integrating both sides from $0$ to $t$ for $u_0=u(0)\in(0,2)$, we have
$$ u\le\frac{2}{1+e^{-2t-c_0}}<2.$$
where $c_0=\ln\frac{u_0}{|2-u_0|}$. For $u>2$, do the same trick we have
$$ u\le\frac{2}{1-e^{-2t-c_0}}.$$
Thus in both cases, we have
$$ \limsup_{t\to\infty}u(t)\le 2$$
where means that the system has a global attractor in $x^2+y^2\le2$. 
