Fix an ODE system $\dot{x} = f(x)$ where $f : \mathbb{R}^n \to \mathbb{R}^n$ is locally Lipschitz. In the case where $x^\ast$ is a global attractor of $f$, it obviously holds true that $x^\ast$ is the unique solution of the system of equations $0 = f(x)$.
I was asking myself whether there are examples that actually show that the converse is not true in general. That is, does anyone know a Lipschitz function $g : \mathbb{R}^n \to \mathbb{R}^n$ where $g(y) = 0$ admits a unique solution $y^\ast$ such that $y^\ast$ is NOT a global attractor of the ODE system $\dot{y} = g(y)$?
Of course. The simplest is for $n=1$: $$\dot{x} = x$$
If you want an example where the unique equilibrium is a local attractor but not a global attractor, you can try e.g.
$$\eqalign{ \dot{x} &= y + (x^2 + y^2 - 1)\; x \cr \dot{y} &= -x + (x^2 + y^2 - 1)\; y\cr}$$
where the basin of attraction of $(0,0)$ is the open unit disk.