Consider the following system of differential equations: $$\dot {x}=y-x+x^3,\qquad \dot{y}=-x.$$ By linearization, it's easy to see that $(0,0)$ is a (nonlinear) sink.
Show that there exists an open connected set $D$ such that if $\phi^t:\mathbb{R}^2\rightarrow \mathbb{R}^2$ is the solution flux, then $\lim\limits_{t\rightarrow +\infty} \phi^t(x_0,y_0) = (0,0)$ iff $(x_0,y_0) \in D$ and such that $\partial D$ is compact and invariant, that is, such that $\phi^t(x_0,y_0) \in \partial D$ for all $t \in \mathbb{R}$ and $(x_0,y_0) \in \partial D$.
Maybe this can be done with a substitution (to show that this system behaves like for instance $\dot {r}=r(r-1)$, $\dot{\theta}=1$) or by finding a function $f \in C^1(\mathbb{R}^2,\mathbb{R})$ such that $(0,0)$ is a minimum for $f$, $\nabla f(z) \cdot \dot{z}<0$ for every $z=(x,y)$ in $D$, and $\nabla f(z) \cdot \dot{z}=0$ for every $z=(x,y)$ in $\partial D$.
The boundary region $\partial D$ is quite visible, and yet it seems that finding a mathematical proof is not so direct...