A centre in the linearization need not be a nonlinear centre ($\sigma (Df(0))=\{i\omega ,-i\omega \}\nRightarrow \text{nonlinear centre}$)

Purely imaginary nonzero eigenvalues make a real planar linearization a centre equilibrium. They do not control the nonlinear radial drift. For example,has the same linearization as a uniform rotation. In polar coordinates, however, $\dot{r}=-r^3$ and $\dot{\theta }=1$. Every sufficiently small nonzero orbit spirals towards the origin, so the origin is an asymptotically stable weak focus, and no nonzero orbit is periodic. Reversing the signs of the cubic terms gives a repelling weak focus. Removing them gives a centre. Thus a nonlinear centre equilibrium requires a further argument, such as a suitable first integral; the imaginary pair alone is insufficient.