

The Bristol Chaotic Pendulum - RiderOfGiraffes
http://maths.straylight.co.uk/archives/301

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lotharbot
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
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.

------
ced
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...](http://en.wikipedia.org/wiki/File:LogisticMap_BifurcationDiagram.png)
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>

------
Marticus
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.

~~~
pbhjpbhj
>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
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.

~~~
lotharbot
> I deviated from that 'few km' by 5,000km.

Reread what you're responding to; he's talking about people staying within
that distance of the surface of the earth, not people staying within that
distance of their home city. The list of people who have been more than ~25 km
away from the surface of the earth is very, very small.

It was a good answer to the previous post: staying within 25 km of the earth's
surface (with a few exceptions) means we're all "predictable" in some sense,
but we're all very unpredictable in other senses -- much like the pendulum is
predictable in some ways and not in others.

------
philh
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.

