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 

$$\begin{aligned} 
  \dot x &amp;= -x -\frac{y}{\ln \left(\sqrt{x^2+y^2}\right)}\\
  \dot y &amp;= -y -\frac{x}{\ln \left(\sqrt{x^2+y^2}\right)}\end{aligned}$$

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.
Many macroeconomists have a somewhat cavalier attitude towards checking the validity of linearization methods, but definitely not all of them. For an example of the approximation issues being taken seriously, see Appendix A.3 on "Log-Linearization and Determinacy of Equilibrium" in the book Interest and Prices by Michael Woodford.