Linearization around an equilibrium point.

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I am trying to understand linearization around an equilibrium point. This is new to me. So I would like to 'see' how it works basically and see how important it is to choose a right equilibrium point Does anyone have a simulation on simulink (or matlab) ? For example, a differential equation equation with nonlinearities that we linearize around an equilibrium point and we can then compare (we plot both at same time linear and nonlinear equation).

I didnt find an example on the internet.

Thank you for your help !