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.
> Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
https://en.m.wikipedia.org/wiki/Nonlinear_system