# What are everyday nonlinear examples?

by UseAsDirected
Tags: chaos, nonlinear, prediction
 HW Helper P: 1,987 You ask for 'everyday examples' and can you identify chaos? I believe it is not easy to identify from observation whether a dynamic is really chaotic, but I will leave this to the experts. In the past engineering sought to avoid it, but these days some work on how to exploit it. The examples often given e.g. above are so simple to enable tractability and not meant to represent reality but rather principle they are called 'toy models'. But at least chaos and non-linearity are fairly new paradigms of dynamics. Before the seventies if they came up they tended to be swept under the carpet. The irregular behaviour of a dripping tap in some conditions of flow is supposed to be chaos. You can at least suspect the erratic and intermittent (equally annoying) behaviour of an old fluorescent lighting tube is chaos. That behind some physical and mental pathologies where the patient is quite unpredictably well and not, or behind some economics phenomena can look like a chaotic dynamic. I would like to hear of better examples.
P: 491
 Quote by Mark44 I think you are confusing linear with deterministic. The population growth of rabbits is NOT linear.
Note the use of the conditional "if".
Mentor
P: 21,304
 Quote by IttyBittyBit Ok, first of all, every single one of you (Mark44, zhentil, UseAsDirected) are making the same, very grave, mistake: assuming that determinism and chaos are incompatible.
Don't include me in this group. I made a point of saying that determinism can be compatible with chaos.
 P: 5,462 CRGreathouse I really think you are trying too hard about this. Linear/ non linear has been around a long time and is understood even at elementary school. Length is linear Area is non linear Volume is non linear But volume can be connected to a linear measure of the quantity of matter by density as weight is linear. What can we pick out of this? Well there is an old idea called proportional to Linear simply means directly proportional to, that is we can convert the proportionality to an equation simply by multiplication by some constant. It is all too easy to mix up ideas from different parts of mathematics and thereby inappropriately apply them. The test of time has worked most of this out of classical mechanics/ mathematics / physics. More recent theory such as Chaos has not yet has that benefit so there is much twaddle about in the subject. A particular faux is to misapply the mathematics of discrete systems to continuous ones, or vice versa. Another is the failure to distinguish between those systems whose mathematics exhibits wildly fluctuating behaviour because they are what is known as ill conditioned or ill posed and those which have an inherent indeterministic element. There are truly random events in the universe. There are also those which are not random but are nevertheless not predictable because of the uncertainty principle. A simple example of a truly random event is the decay of a single atom of uranium. Statistics will tell us how many atoms on average will decay in any given time period, but it cannot tell us when a particular atom will do so. It cannot even prove that when there is only one left it will actually decay at all. Statistics again can tell us what percentage of molecules will be involved in a particular chemical reaction, but cannot tell us for certain which ones will be included and which ones will be left out. Finally if you want some circus for your target audience I suggest you read the excellent Oxford University Book From Calculus to Chaos by David Acheson.
HW Helper
P: 3,684
 Quote by Studiot Length is linear Area is non linear Volume is non linear
You can't say that in such generality! Volume varies nonlinearly with surface area, for similar shapes. But it varies linearly with, say, mass for objects of uniform density. It's easy enough to give counterexamples for each of your statements.
 P: 5,462 CRGreathouse I apologise for misplacing you as the OP. My comments were meant as a genuine reply to the original post, whgere they make more sense. I don't understand your desire to generate 'counterexamples' - I am trying to offer suitable examples for what I understand to be gifted if difficult youngsters. In particular area as an xy product is definitely non linear. All I am saying is that we all reasily handle non linearity in our everyday lives, so we shouldn't be frightened of it or awed by it.
P: 159
 Don't include me in this group. I made a point of saying that determinism can be compatible with chaos.

That being said, I think it's delete/ignore time for this thread!
HW Helper
P: 3,684
 Quote by Studiot I don't understand your desire to generate 'counterexamples' - I am trying to offer suitable examples for what I understand to be gifted if difficult youngsters.
My desire is to ensure that what examples we give are correct.

 Quote by Studiot In particular area as an xy product is definitely non linear.
The area of a rectangle with height 1 and width w is a linear function of the width.
 P: 14 I'll tell you straightaway that I now see I am not the only one not understanding nonlinear systems. >Quote: That said, I think it's delete/ignore time for this thread!" is a bit presumptuous, though I am not aiming this at the above poster. One response sifted from all these is dishonest, copied from a part of BBC4's Chaos" but with an air of authority here, another convoluted, at first appreciating the 'everyday example' attempt, only to defer on it and splice it with a book review (of all things), and one or two just honestly wrong. Actually, I appreciate the honestly wrong ones because it is a chance to flush out some bad ideas, get a conversation going, and reflect on a habit of mind. My original post lead soon after to the main question, and I asked it perhaps two more times about whether there is a scientific concept to non-linearity, if at all. This all started when I was in a meeting last Thursday about teaching a nonlinear workshop to gifted physics students and a faculty member responded to an education promoter, but I can't stop to get the exactness, to the effect, Wait, what? But there are no science concepts to teach. It's just math …notwithstanding we are entering into an age of `hyphenated' subjects…I think you should take this to the math school." Hmmm...interesting, I thought. I was hoping to get a better sense here and think, I tossed this message out to a web-board the other day and public opinion claims there is a scientific concept to teach in nonlinear systems. And it is…" So, never mind, I suppose. Thanks, -E
 P: 10 WHY is there so much confusion over this topic? OP asked how to tell if a physical system or set of equations is nonlinear. The answer is, a physical system is said to be nonlinear if the EQUATIONS governing it are not linear. i.e. the differential equations involve terms which are not scalar multiples of the unknown and its derivatives. For example, y' +2xy = e^x is linear, but y'^2 +2xy = e^y is a nonlinear differential equation. So you can tell immediately by looking at the equations if a system is linear by checking if the equation contains any nonlinear functions of the solution or its derivatives. Chaos and all that is a specific trait of some nonlinear equations. But in the full generality, you can't say much about the solutions to nonlinear equations.

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