In my life I hear/read this statement a lot:
A non-linear equation or theory leads to self-interactions.
For example in GR, we say that gravity can interact with itself because it is non-linear. For some reason I always assumed it was correct. However now I think about it, I can not see a clear reason in the maths for why this statement holds. Can someone help me out? :D
Edit$_1$: As Vadim pointed out. The statement should be the other way around.
A self interacting physical system leads to non-linear equations.
Edit$_2$: The questions is beautifully answered by @gandalf61 for 2 variable system. However, still do not really understand what is going on for 1 variable system, e.g. in general relativity. Could someone maybe also give an example there? Thank you in advance. :D
In the comments on the answer of @gandalf61, you will also find the answer of edit$_2$.