TLDR: The "most addictive theorem" is the Hartman–Grobman theorem which states that under some very specific and technical conditions non-linear system near equilibrium can be described with a linear system.
Imagine you're looking at a ball rolling around on a complicated curved surface defined by differential (non-linear) equations, which can be a tricky system to analyze.
If the ball is near some sort of a saddle equilibrium point, the theorem says that you can simplify things by flattening out a small patch of the surface if you are (very) near that point.
This is a lot easier to analyze using simple linear algebra tools, and still gives good results for predicting what happens next.
Generally it's a pretty basic concept in analyzing any system, 1st step is to use a constant, if it doesn't fit then a linear model, etc. Can use different approximations depending on the circumstance, e.g. you can approximate sin(x) with x around 0...
Is there something special to it in this particular case?
Hartman-Grobman isn't so much about approximation error by choosing different simplifying models. It's about being able to do some analysis on the linear approximation with the results carrying over to the original non-linear system. In particular, it lets you categorize stationary points of a non-linear dynamical system, using only the linear approximation at the fixed points, which is awesome since analyzing linear systems is super easy.
My take on this theorem is that it is most interesting to know when the topological equivalence doesn't hold, i.e. linear approx of equilibrium point can be poor when there is the possibility of a limit cycle oscillation.
Not sure what your issue with "hard to analyze" is. When compared to linear systems, nonlinear dynamical systems are hard to analyze.
> Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
I think you are just reading too much into some colloquial English. If I say that "Aunt Elsie coming to dinner means trouble", I don't literally mean that "Aunt Elsie coming for dinner" is a synonym for any kind of trouble. It is clear just from the context of the sentence you quoted that this non-literal meaning is intended, since "difficult to analyse" is very obviously not a mathematical definition. And in case this was not clear to anyone - which I already find surprising - they inserted the word "basically" which is another colloquialism that can mean "not literally" (depending on the context).
That is not the meaning of the term. However, if you are to describe what the big deal is about finding ways to analyze nonlinear systems to lay people then what better description is there?
First line of the Wikipedia page is fine: "In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input."
Scientific American's typical reader surely understands a) the concept of a function b) the concept of proportionality.
It's fine to say that nonlinear systems are difficult to analyze, or have a reputation for being difficult to analyze. But it's bad practice for a science communicator even to run the risk of giving people the impression that the difficulty is part of the meaning (definition) of the word.
Did the article change? The rest of the “hard to analyze” sentence you quoted above is “linear systems respond proportionally to changes in variables, whereas nonlinear systems have more complicated relationships.“
The line you quote from Wikipedia is not the definition of a nonlinear system.
EDIT: The definition depends on the area of math you are working in. The definition of a nonlinear system of differential equations is different than the definition of a nonlinear system of polynomial equations in commutative algebra. As far as the article in question it's an accurate description of the problem and why it's a big deal.
Some people are stuck in seeing everything gradual. They tend to discount systems flipping and long tail events.
Then there a few who see mostly nonlinear systems and neglect the spoils that lay in linear analysis of a localized space.
And on microscopic level there is the weirdness of quantum effects.
Knowing into which toolbox to reach is not always clear.