Can a gradient system such as
x' = $\frac{dH}{dx}$
y' = $\frac{dH}{dy}$
where H $\in C^2(\mathbb{R})$
have a focus on a critical point? If so, could anyone give me a example?
I know that the linear part of a gradient system cannot have a focus on a critical point, but does it mean that the non-linear system cannot have a critical point either? For instance, is it possible that the linear system has a node on the origin but the non-linear system has a focus?