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To quote Physics.SE tag definition:

The term non-linear or nonlinear has several definitions but is generally used to describe a system that cannot be approximated by a superposition principle or perturbative approach.

...and one could add that these systems are usually described by nonlinear differential equations (which prevents the superposition etc.). That's clear.

Now, I need to explain typical features of this behavior without the math. Please note that "explain" does not mean "define", that would be an obvious dead end.

Since this is very broad, let me narrow it down. I need this for a popular lecture about fluid dynamics and turbulence. My aim is not to use sentences like "We have here a highly non-linear flow" without explaining or exemplifying the term.

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  • $\begingroup$ Well, when they exist, I would think conserved quantities like vorticity are what make non-linear systems interesting, and control most of the qualitative behavior. Also, stability analysis is useful for determining when you can expect perturbative approximations from exact solutions to fail. $\endgroup$ – TotallyRhombus Jul 11 '16 at 14:56
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    $\begingroup$ One thing you should mention in your lecture is that all physical systems are nonlinear in some respect, even those that may appear to be linear. You may just need to look closer, or drive the system harder to expose it. Being linear is an idealistic model of nature. Nature is not models. Models are an invention of man. $\endgroup$ – docscience Jul 12 '16 at 2:15
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First, give an illustration of a linear system. You know you have to eat healthy, so you are prepared to pay 50 cents for an apple. But the next day, if apple prices double, you will forget the health benefits.

Now a non linear system, because it's based on people's reaction to supply and demand.

An everyday example based on the Uber taxis system. Uber connects passages seeking rides home with drivers looking to make money from renting out their cars. Rates increase when enough drivers feel they made enough money, resulting in a car shortage as demand exceeds supply.

Its relativity easy to get a car at some late hour such as 4am, but there are also peak times as offices close at 6pm. Putting the price up then does two things, gets more Uber cars out on the road and allows them to make more money.

But as more people use Uber, they get used to the surge pricing system and time their journeys to avoid it. It's not as straightforward as the linear response to the price of apples. The response of people is now involved and alters the linear pricing system as more people become aware of the surge price increase, whereas other people may have to pay the extra money, because they must make the journey.

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