Well, subject to certain constraints.

You aren't guaranteed that your equilibria behave the same in the linearization of a nonlinear system if your Jacobian has any eigenvalues with real part of 0.

https://en.wikipedia.org/wiki/Hartman%E2%80%93Grobman_theore...

It's amazing how many ideas I've seen being "rediscovered" these days from control theory.

Wait isn't that just an application of Taylor expansion which analysis has known about for several hundreds of years?

Taylor is only for some functions, must be continuos and differentiable to n grades. In control theory you break the functions, so discontinuities can also be considered.

Alternatively, anything is linear if you add enough dimensions.