The Bristol Chaotic Pendulum

http://maths.straylight.co.uk/archives/301

24 13 16 years ago

maths.straylight.co.uk

lotharbot3• 16 years ago

What makes it "chaotic" isn't just that it'll turn right if you start it leaning a tiny bit right and left if you start it leaning a tiny bit left. What makes it chaotic is that, if you start it in position X, it might make the series of turns RRLRLLLRL... while if you start it in position X+epsilon, it might make the series of turns RRLRLLLLR... or LLRLRRRLR... or some other completely different sequence. It may even make the first hundred turns exactly the same, and then diverge at turn #101.

In other words, any tiny change in the initial conditions, even if it doesn't change the first turn, can change the pattern of turns farther down the line. In chaos, sensitive dependence on initial conditions doesn't just happen at a single point, but throughout the system. It's not just a matter of making or missing the train, but of talking to different people or having slightly different mannerisms because you were 3 minutes and 2.8 seconds early instead of 3 minutes and 2.7 seconds early, and those tiny changes eventually adding up to lead you to Boise instead of Miami.

jsomers• 16 years ago

Nicely put. I often consider this problem when I think counterfactually: "What would have happened if I missed the train? Where would I be right now if I hadn't lost my pencil two days ago?"

I have an instinct to just dismiss this kind of musing, because I think to myself that even the smallest changes propagate in wildly unpredictable ways, with potentially vast consequences -- think "the butterfly effect"-- and so it's pointless to even try to reason counterfactually.

But of course it's not pointless, and our minds do this naturally all the time: holding everything else equal, we're constantly twisting this knob and that one, imagining all kinds of possible worlds, exploring the space of what could have been.

philh• 16 years ago

I remember seeing that when (I think) I was sat in the church for a school service one year. I'd forgotten all about it, thanks for the reminder.

Marticus2• 16 years ago

Well, from looking at it, it's actually very predictable.

Once one arm crosses what I'll call the lower bound (at roughly 80+ degrees from the bar being parallel to the ground), the water empties so rapidly that the tube filling up is not nearly fast enough, so the pendulum tilts back to the other side.

There is also an "upper" bound on each side, which is during the filling phase. If the arm doesn't switch sides and "stops" roughly within, say, 15 degrees from parallel, the water will fill up to the point that it will become so heavy at that side that it will tip, satisfy the lower bound, and switch sides.

Otherwise the "current" side will oscillate between those two bounds until it breaches (usually) the upper bound.

RiderOfGiraffes• 16 years ago

I think you'll find that it varies according to how fast it's going when it gets to one of the bounds that you think you've observed. It's not as predicatable as you seem to think.

Marticus2• 16 years ago

Not really, because it has hard boundaries, that is to say it can never "level" at perfectly parallel to the ground (obviously), and cannot achieve perfectly perpendicular alignment to the ground.

Speeds with such limited motion are actually quite limited themselves along the circle.

njharman• 16 years ago

> Well, from looking at it, it's actually very predictable.

Yes, that's it of course! you have single handedly out proven hundreds of man years of scientific study from watching a 5min video!

pbhjpbhj• 16 years ago

>Well, from looking at it, it's actually very predictable.

People are very predictable too. On the whole they walk on the surface of the Earth and don't deviate from it more than a few km.

electromagnetic• 16 years ago

I deviated from that 'few km' by 5,000km. I also regularly deviated a several hundred km to France with my family. My in-laws regularly deviate a thousand km to the east coast of Canada. One of my bosses regularly deviates to the southern USA.

It's a very misleading assumption to say people don't deviate far from where they live. I know here in Canada it's not only common, but considered abnormal if people don't regularly go to cottage country, frequently travelling a minimum of 100km to get there.

Just because city-dwellers naively believe the city has everything they need, doesn't mean the 4/5ths of the population that permanently live outside of large cities behave like those in the cities.

Even when I lived in the UK, it was rare to know people who didn't travel far frequently.

lotharbot3• 16 years ago

It depends on what you mean by "predictable". Chaos does not necessarily mean you can't tell where something is going next. It means you can't reliably predict it over the long term.

If you were to watch the pendulum, you might be able to reliably make statements like "it's going to tip to the right in a few seconds" (particularly when it's near one of the bounds), but you won't be able to reliably predict which direction it will be tipped 3 minutes from now, or whether it will spend 8 seconds or 28 seconds tipped to the right, or whether its next 10 motions will be RLLRLRRRLL or RRRLLRLLRL. Any tiny error you make in measuring or predicting will be magnified over and over until the error is bigger than the signal/calculations.

We're all familiar with another chaotic phenomenon: weather. Tomorrow's temperature and prevailing atmospheric conditions can be predicted fairly well, but you can't make reliable weather projections more than a few days out. It's intrinsic to the system; a tiny effect like a butterfly flapping its wings half a world away is enough to make your predictions worthless a couple weeks out.

ced• 16 years ago

Chaotic behavior is surprisingly simple to generate. The equation:

x[n+1] = alpha * x[n] * (x[n] - 1)

is chaotic for alpha > 3.56. In other words, if you run this Python code:

   def chaos(alpha):
       x = 0.5 # The "initial condition"
       l = []
       for i in range(100):
           x = alpha * x * (1 - x)  # The logistic equation
           l.append(x)
       return l

then

chaos(2.6) yields [...., 0.615, 0.615, 0.615, 0.615, 0.615, 0.615, ....]

chaos(3.2) yields [...., 0.513, 0.799, 0.513, 0.799, 0.513, 0.799, ....] (a 2-periodic sequence)

chaos(3.8) yields [...., 0.649, 0.865, 0.443, 0.937, 0.221, 0.656, ...] (non-periodic --- chaos!)

When we say that chaotic systems are unpredictable, it means that small differences in the initial conditions get amplified. So, if I replace x = 0.5 with x = 0.50001, then the result of chaos(2.6) will be (almost) the same because it's not chaotic, but the result of chaos(3.8) will be completely different.

The diagram at http://en.wikipedia.org/wiki/File:LogisticMap_BifurcationDia... is drawn entirely using the equation above, and it's the simplest example of funky mathematics that I know of. See also http://en.wikipedia.org/wiki/Logistic_map

The Bristol Chaotic Pendulum

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