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