The Bristol Chaotic Pendulum
maths.straylight.co.uk
maths.straylight.co.uk
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.
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.
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
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
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.
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.
Yes, that's it of course! you have single handedly out proven hundreds of man years of scientific study from watching a 5min video!
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.
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.
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.
Speeds with such limited motion are actually quite limited themselves along the circle.