Last week I made the mistake of trying to feed applesauce and

yogurt to my 19-month old daughter in a single sitting, and letting

her use the spoon and handle the small open bowls by herself. As I

cleaned up the resulting disaster, it occurred to me that I had yet to

do a podcast on chaos theory. In addition, this topic has been in the

news lately due to the death of one of its modern founders, American

climate scientist Edward Lorenz.

Not a mathematician by trade, Lorenz got involved in this area

back in 1960, when he was implementing early computer models of how

air moves around in the atmosphere. He was hastily trying to rerun

some calculations, and the second time around, to save time, he

rounded off some input values to only a couple of decimal digits.

Percentage-wise, this was a miniscule change in the values. But he

was surprised to see that when he reran his simulation, he got

completely different results. This led to a fundamental insight, that

certain classes of functions, now known as chaotic functions, can

generate widely varying results due to a very slight change in input

conditions. This leads to the famous "butterfly effect", where in

certain models, the flapping of a butterfly's wings in Texas can

eventually make the difference in whether a tornado strikes Brazil.

We should also point out that while Lorenz founded modern chaos

theory, the general concepts were known and discussed by

mathematicians since Poincare' in 1890. But the existence of modern

computers is what made its study practical.

As a simple example, let's look at the function f(x) = x squared +

1/4, and see what happens when we continually iterate it, or apply the

function again to its result. If we start with 1/2, you can see that

the function yields itself, so we just repeatedly get 1/2 forever. If

you apply 0, you first get 1/4, then .3125, then approximately

.3477... and if you repeat enough times, you'll find the result gets

closer and closer to 1/2. Try a few other numbers between 0 and 1/2,

and you'll see similar results. But now try a number just slightly

higher than 1/2, like .51. It's only off by 2%, so you should get to a

result that's pretty close, right? Wrong! You will see that the

results of this process veer off to infinity for any number greater

than 1/2. It turns out that the range of numbers with absolute value

less than or equal to 1/2 all lead to the result of 1/2, which is

known as an "attractor". For any number greater than 1/2, this

process leads to infinity! If you follow the references in the show

notes, you'll also see much cooler multidimensional function plots

where the attractor is a solid in three-space, a set of

three-dimensional values that various ranges of inputs will lead to.

Lorenz discovered one such object, known as the "Lorenz attractor".

You should see that while chaotic functions display fundamental

changes in output based on slight changes in input conditions, they

are still fully determinstic, without any true random element. The

results may seem random to us, but that is entirely a result of not

having precise enough knowledge of the correct input conditions. If I

had precise enough knowledge of the workings of my daughter's brain,

and measured to the nanometer exactly where I initially placed the

bowls, perhaps I could figure out exactly where each blob of yogurt

or applesauce would end up on her and my clothing. But unfortunately,

at my knowledge level, I am doomed to have to guess the results and

deal with the apparent randomness.

So, what does chaos theory mean in practice? The fundamental

insight is that even if a process is fully determinstic and

scientifically modeled, it may be impossible for us to accurately know

the result if there is even the slightest uncertaintly about the

inputs. When someone presents you with a computer simulation claiming

to prove something, you first have to ask whether they are analyzing a

chaotic function, and if so, how precise do they really have the

starting conditions. So any scientist in a discipline that uses

computer simulations and modelling must be aware of and consider chaos

theory. In part, this provides an explanation for why TV weather

prediction is so often inaccurate; no matter how good the models are,

there is never truly enough knowledge of the inputs to guarantee a

correct result.

And this has been your math mutation for today.

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