In economic modelling, how are we sure the linearized equilibria behave similarly to original equilibria?

The following problem is in the context of continuous time, although I suspect something could be said about discrete time also.

Let's assume that we have the following equation:

$$\dot x=f(x)$$

where $x(t)\in \mathbb{R}^n$, for $t\in I\subset \mathbb{R}$.

I'm reading a book on Ordinary Differential Equations. In this book, they state that one should verify some conditions on $f$, usually $f\in C^2(E\subset \mathbb{R}^n)$, to be sure that the linearization of $f$ behaves in a qualitatively similar manner (ensures existence of a $C^1$-diffeomorphism).

For example, consider $$\dot x=-x-\frac{y}{ln(\sqrt{x^2+y^2})},\dot y=-y-\frac{x}{ln(\sqrt{x^2+y^2})}$$

with $f(0)=0$. This $f$ is only $C^1$ not $C^2$(transform to polar coordinates$(\theta,r)$, and then see $\frac{d^2 \dot \theta}{dr^2}$ tends to infinity as $r$ goes to zero). The linearization gives a stable node, but according to a more general definition, we have a stable focus for the original non-linear system.

However, whenever I see any kind of linearisation being done (DSGE or Growth Theory), not once do I see any concern related to this... Maybe, there's something, which I'm missing, that makes economic diff. equations already satisfy this. Or am I wrong, and this simply isn't, and shouldn't be a concern to theoretical macroeconomists?

Any help would be appreciated.