What are everyday ``nonlinear" examples? Hello! Is there a simple way to identify a nonlinear equation or physical system by looking at it? I have sifted through material about unpredictability, chaos, fractals, and the other buzzwords encompassing ``nonlinear systems", and have glossed over mathematical explanations covered in Wiki articles, but do not seem to understand how to identify an algebraic nonlinear example other than ``variable cannot be separated", ``superimposed," is "non-homogenous". I am seeking a basic explanation for rather young kids in a gifted physics program. For example, is an exponential, logarithm, root, or quadratic nonlinear as they do not conform to ``lines"? (Notwithstanding that I can just plot it against two logs to get a line.) And also, the above examples are ``predictable," no? Are they still nonlinear? Does nonlinearity imply that despite knowing the initial conditions the outcome cannot be predicted? (I solved the intersection of a quadratic and a linear equation, found two points, and am concluding the system is ``nonlinear" because the ``nonlinear" shape of x^2 (parabolic) causes the equation to be a ``system" of solutions (more than one point satisfies the bounds). I am seeking an elementary school explanation and basic examples. Thanks, -E
Re: What are everyday ``nonlinear" examples? In my mind most things are nonlinear. Treating things (within a fixed range) as though they were linear is a mathematical trick that lets us work with complicated things as if they were simple. Your examples are all simple, predictable, and nonlinear. No, that's "chaotic", not "nonlinear". But multivariate nonlinear systems are often chaotic.
Re: What are everyday ``nonlinear" examples? Thank you for the response. I get it now that nonlinear systems can be simple and deterministic. And, does ``sensitivity" to initial conditions imply chaos? I read about these buzzwords and that due to such and such sensitivity hither tither system is ``chaotic" or not-deterministic. Do I conflate the two, chaos and non-determinism? I am trying to build up a catalog of understanding. Thanks again. -E
Re: What are everyday ``nonlinear" examples? No, sensitivity to initial conditions is not enough for chaos. To have chaos, you also need to visit every point in the phase space arbitrarily closely an infinite number of times, at any large time frame (this is just a rough explanation; to really define chaos you need to get nitty gritty with higher mathematics). What does this mean? Consider the function x^n as n goes to infinity. This function is sensitive to initial conditions, since starting with 1 it will just give 1, but start at any number greater than 1, say 1.000000000001, and it will give infinity. So it's really sensitive to initial conditions. But it's not chaotic; it's simple and predictable. The logistic map is a really simple (and good) introduction to chaos, you might want to read about it if you haven't already: http://en.wikipedia.org/wiki/Logistic_map And non-determinism doesn't imply chaos either. A random (not pseudorandom!) sequence of numbers is non-deterministic, but non-chaotic.
Re: What are everyday ``nonlinear" examples? The Mandelbrot set is determined by a very simple nonlinear equation: z = z^{2} + c, where z and c are complex numbers. I might be wrong, but I think that the Mandelbrot set is deterministic in the sense that given a complex number you can determine whether it is in the set or not, but very small changes in input values lead to very different outcomes, so the set is very sensitive to changes in initial conditions, hence chaotic. There's a nice animation about halfway down the page at http://en.wikipedia.org/wiki/Mandelbrot_set, in the section titled Zoom animation. Another example of a simple, non-linear equation with chaotic behavior is in the Bifurcation topic here http://mathworld.wolfram.com/Bifurcation.html. The graph is generated by various values of r in the equation x_{n} = rx_{n - 1}(1 - x_{n - 1}). If you look at the graph, the two left-most red lines are at r = 3.44 and x = .44, x = .85. Substituting .44 for x_{0} in the equation above gives x_{1} = 3.44*.44*(1 - .44) ~ .85. Substituting this value in the equation gives x_{1} = 3.44*.85(1 - .85) ~ .44. Varying r by a little bit causes a small variation in the output values, but varying r by a little more causes bifurcations at around r = 3.45, and more at around 3.545, but the system really goes bonkers at r = 3.57 or so.
Re: What are everyday ``nonlinear" examples? Here's a way to think about it: suppose you're measuring the population of bunnies. You know that right now there are 100 bunnies and that the population is growing at the rate of four bunnies per week. If this data allows you to determine the population at all future times, the system is linear. Otherwise, it's nonlinear. More generally, a system is linear if and only if knowledge of the function and its derivative at any point allows you to completely determine the function.
Re: What are everyday ``nonlinear" examples? I think you are confusing linear with deterministic. The population growth of rabbits is NOT linear.
Re: What are everyday ``nonlinear" examples? Thanks for the responses. To Ittybitty; hi, you wrote: >This function is sensitive to initial conditions, since starting with 1 it will just give 1, but start >at any number greater than 1, say 1.000000000001, and it will give infinity. So it's really >sensitive to initial conditions. But it's not chaotic; it's simple and predictable. What is ``sensitive" about it? I raised a larger number, 1.00001 to 50 and is 1.0005, 1.00001^500 = 1.005, and 1.00001^5,000 = 1.051. It doesn't seem sensitive at all. I have another, perhaps more pragmatic physics education questions. Q.) Are their key *science* concepts to non-linearity? Or, is it just mathematics? Thanks, -E
Re: What are everyday ``nonlinear" examples? Here's an example of a reasonably chaotic (though deterministic) system I worked with a few years back. Take a starting value, say 1. Repeatedly apply the tangent function until it is in a given range, say [319, 320).
Re: What are everyday ``nonlinear" examples? I just did it in Excel and it's quite cool! So, it is chaotic yet ordered?
Re: What are everyday ``nonlinear" examples? Ittybitty said x^n as n goes to infinity. 50, 500, and 5000 are insignificantly small in comparison to infinity.
Re: What are everyday ``nonlinear" examples? 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. In fact, one of the defining marks of chaos is that it has to be deterministic. let me repeat: chaos is always deterministic. In fact, that's why we call it deterministic chaos. Non-deterministic 'chaos' is just randomness. And randomness is not chaos, it's just randomness. Like the throw of a dice or what age you will die at. This is actually a common misconception. Now another misconception is about linear vs. nonlinear systems. The problem arises from the fact that 'linear' is typically taken to loosely mean 'easy' in engineering courses. The word 'linear' does not have a mathematical definition (perhaps the closest thing to mathematical definition would be that a linear system is something that satisfies the superposition principle), but things like 'linear transforms on vector spaces' or 'systems of linear equations' do. Thus it is wrong, in my opinion, to make sweeping assertions about something being 'linear' or not; we have to study the concept in the context it is meant to be studied. Yup.
Re: What are everyday ``nonlinear" examples? In this example, throwing of die is random? I think the fall of die is calculable, provided all the minutest details are known. And, what is the relationship between ``determinism" and ``predictability"? Thanks, -E
Re: What are everyday ``nonlinear" examples? Well it is possible for something to be deterministic, yet not possible for us to be able to predict it. In chaos, usually you have this process where, as you progress forwards in time, the exact details of the initial conditions become more and more important. Take the weather for example (a chaotic process). We have the equations to model it, it's just that we can never know all the initial conditions with 100% accuracy. Thus our prediction ability is limited to just a few days in the future. We can never hope to track every single child across the world blowing bubbles into the wind, for example. You know about the butterfly effect right? That a single butterfly, flapping it's wings in, say, china, can lead to the difference between a hurricane striking or not striking a city on the coast of the US. The interesting thing about chaos is that this is guaranteed to happen; a flap of a butterfly's wings will, without a doubt, be translated into a storm being created or not. But with randomness, the picture is different. If something is truly random, we can't even predict it on the shortest time-scale. You are right, throwing a dice is not in fact random. It's actually very hard to construct a perfectly random sequence. Some people have done this with quantum devices, but even those are based upon the assumption that the quantum world is truly random, something that has not been proven.
Re: What are everyday ``nonlinear" examples? The butterfly's act of flapping is guaranteed to create a storm or not? Is ``or not" part a typo? Has it been shown experimentally that a butterfly's flapping guarantees the initiation of a storm? If we cannot micro-analyse the initial conditions of a system, how can we possibly demonstrate this? I still wonder about this pragmatic issue: Q.) What is the key scientific concept in non-linearity[, if there is one]? Or, does it belong to mathematics? Thanks, -E
Re: What are everyday ``nonlinear" examples? No it's not a typo, I just explained it really badly. What I meant to say is: To model the state of a system as it evolves in time, we eventually need to know finer and finer details about the initial conditions, with no limit to how far we have to go. This has been proven with a mathematical analysis of the subject of chaos, btw. In fact, it is one of the defining characteristics of chaos. A system that 'forgives' perturbations in the initial conditions smaller than a certain threshold is not deemed chaotic. About linear/nonlinear systems: you might want to start with researching the superposition principle.
Re: What are everyday ``nonlinear" examples? The questions really are several, I will divide my answers. Although to CRGreathouse's mind most things are nonlinear to my mind enough things are linear enough for linearity to be useful and essential. That is if they are a bit nonlinear, e.g. if the restoring force in an oscillation curves a bit you can still treat it acceptably depending on accuracy required as linear and get useful results. The concepts are still useful. The qualitative behaviour carries over. For instance even the simple pendulum never has a linear restoring force, it depends on sine of displacement not displacement. Nevertheless even when not quite right the period is still independent, exactly or approximately I don't remember, of maximum or initial displacement i.e. also amplitude. Therefore this is not essentially nonlinear. It is linear for small displacements still. 'Nonlinear science' - think its practitioners think this way - is when you have essentially qualitatively different behaviour from the linear. Thus the nonlinear version of the simple pendulum would be the grandfather clock. Its final period and trajectory or amplitude is independent of initial displacement - called a 'limit cycle'. That is a qualitative difference between the two dynamics.
Re: What are everyday ``nonlinear" examples? 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!
Re: What are everyday ``nonlinear" examples? 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.