# 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 &= -x -\frac{y}{\ln \left(\sqrt{x^2+y^2}\right)}\\ \dot y &= -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.

• @RodrigodeAzevedo the same technique is applied to DSGE models. May 7, 2020 at 13:02