I am quite sure that most phenomena in nature are nonlinear; in any case, there is very little motivation that it should be linear. We can dream of linear-elastic phenomena, but it is most often a convenient assumption when we perform sensitivity analysis only.

I am quite sure that most phenomena in nature are nonlinear; in any case, there is very little motivation that it should be linear. We can dream of linear-elastic phenomena, but it is most often a convenient assumption when we perform sensitivity analysis only.

Actually, the generally accepted perception is that many shapes in nature, if not all, are irregular and rugged, or chaotic (nature is fractal).

What do you even mean by 'nature', as it is not even observer-independent? Article Experimental test of local observer independence

Let's return to the basics. In uniform motion, the distance travelled grows linearly with time and with velocity. One might object that there is no true uniform motion in nature (although some examples get pretty close to it) and that this is basically the definition of velocity. No such objections apply to the second Newton's law, which states that the force grows linearly with mass and linearly with acceleration. It holds true if we neglect relativistic effects - if we don't, we'll need a different example again.

If you are looking for examples outside of basic physics and closer to ecology and natural resources, the amount of grass grown is linearly proportional to the size of the meadow, the amount of grass eaten is proportional to the number of deer eating it, the total cellulose content of the grass is proportional to the amount of grass. When pumping water, the amount of water pumped is proportional to time, to the power of the pump, or to the number of pumps. And so on.

To conclude, I can't think of an example that is particularly exact or insightful or that has a deep important meaning. The given examples may serve to illustrate how we often assume linearity and use it in our research and in daily life, and how much harder our lives would be if we didn't do it.

Dear Matija Perne ,

I think you are not answering the original question. Is linearity assumption useful and actually accurate in many applications? Absolutely. But "do we actually have linear system in nature" or are they "an approximation of nature"?

None of the examples you provided are linear. Let us take your grass and deer example: what about the negative numbers of deer? Or grass? Or even not an integer number of deer (they are quantized, as far as I can tell)? Or even outside of the mathematical definition of linearity, what happens when the number of deer becomes too large to be supported by the meadow of a fixed size - then the 'eating rate' will be capped at some point. Does the postulated proportionality hold at any point in time, or averaged over some period of time?

Ggiven that the absolute majority of "things" in the universe (mass/energy etc) are quantized, the existence of anything "truly linear" seems to be impossible. Maybe one can do something "truly linear" with time or gravity, but then one gets into things like "what happened before the big bang" - or even "what is time any ways"...

Currently teaching Linear Algebra to Sustainable Design students, I am also on the search for real-life motivational examples the Linear Algebra machinery I teach. I guess any system in the physical world will have features or components your can point to and say: "Because of this feature, the system is not linear in the strictly mathematical sense". But we can also find linear approximations that for the given task at hand is in every practical sense sufficiently linear so that we can make our predictions to our desired accuracy with a linear model. I guess this is the point Anton is making above.

But I also want to make another point here. You talk about linear vs complex. Linear systems cannot exhibit chaos, true, but non-linear systems can also be quite boring and practically very predictable. For instance the 3-body problem is chaotic, but if one of the bodies is very heavy and another very light and they are sufficiently far apart (like with the Sun, Jupiter and Earth) the problem becomes regular and predictable in spite of it's fundamental chaotic nature. My point is that complexity in a system is not a matter of linear vs. nonlinear. Rather it is a matter of "sensitivity to initial conditions" of the governing equations in the part of the parameter space where the system resides.

Dear Lucia: Linearity is easy to understand and teach - the price we need to pay for such simplification is, sometimes, too high, rendering the models unrealistic. We pay a price for all models (e.g., mathematical or statistical ones), but at least we try to get closer to -what we think- represents the real phenomenon.