What are everyday ``nonlinear" examples?

epenguin

For students I would have recommended the system mentioned by Mark44. If you have just a hand graphics scientific calculator, and a very simple programme, they can have fun and surprises with that system.

A best short account and introduction to it in a few pages is probably still "Simple Mathematical Models with very complicated dynamics" by Robert May, Nature, 1976

The kind of diag. on the cover of the book below is key to throwing light on the strange behaviours. It is a look behind the iteration - I think you can see what it is. Here r is above 1 and below 3 I think and you see x homes in on a single stationary point. (For r above 2 it homes in in the oscillatory manner shown.)

One of the concepts to come out of chaos studies with this simple system was universality. That is Mark's illustration is with the simplest formula but it didn't much matter what the formula is as long as it has an extremum basically - it can even be a 'tent' - a straight line up and then down like ^. Even more surprising, chaos set on for the same ('universal') value of the controlling parameter (height of maximum) whatever the function chosen, and qualitatively the approach to it via period doublings was the same for different functions! Students can have fun with these too.




For books the above one by Holmgren is at least short. I found it disappointing - it told me everything I had already worked out and nothing I wanted to know - e.g. proper explanation of universality. The combination of things it expected you to know (topological terminology) and the elementary things it thought needed lengthy explanation were to me a bit disconcerting, but basically the math is elementary.

For the teacher a longer and wider and better book - but as I say longer - is 'Chaos and Fractals' by Pietgen Jurgens and Saupe. High school math (in Europe) is enough for it - there is more but it is explained.

Those 3 refs should keep you busy quite some time!

epenguin

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.

zhentil

Note the use of the conditional "if".

Mark44

Don't include me in this group. I made a point of saying that determinism can be compatible with chaos.

Studiot

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.

CRGreathouse

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.

Studiot

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.

IttyBittyBit

Hmm. Apparently I misread your comment about mandelbrot sets. Sorry.

That being said, I think it's delete/ignore time for this thread!

CRGreathouse

My desire is to ensure that what examples we give are correct.

The area of a rectangle with height 1 and width w is a linear function of the width.

UseAsDirected

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

colldood

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.