If someone showed you a swinging pendulum—like this and asked you to calculate the pendulum’s
position at some point in the future, you might think it’d be relatively simple.
We should be able to predict that with the known laws of motion, right?
But this calculation is actually so complicated that it requires a relatively powerful computer.
And even if you did do this calculation, and then are offered slightly different initial
conditions, you would think that the output might also only be slightly different.
But instead, you get....chaos. Specifically, deterministic chaos.
See, as humans, we’re always trying to know more about how our world works, so we make
models.
For example, we have a bowling ball.
We know the weight of the bowling ball, the effect of gravity, the density of the air,
and the height of the balcony from which the bowling ball is to be dropped, and we can
put all those things together into a system of equations that we could use to tell us
things like how fast the bowling ball would fall, or the force with which it would impact
the ground.
That means we can also reasonably predict what might happen if any of those variables
were to change.
This is a deterministic system—the behavior of certain variables is determined by their
known characteristics. But not a lot of the world is like the bowling ball example.
It’s a whole lot messier.
Take the weather.
Just think about all the things that go into making weather—temperature, humidity, wind
strength and direction, rotation of the earth, I mean the list is exhaustive.
We do have models for how all these many variables behave, but a perfect weather prediction would
require highly accurate measurements of all of the contributing variables over every square
inch of the piece of atmosphere we’re looking at. In a system like this, little measurement
errors can result in HUGE fluctuations of our calculated result.
Tiny changes of the input mean a LARGE variation of the output. Now don’t get me wrong—the
system is still deterministic.
The variables behave the way we expect them to based on their physical properties, so
it’s quite different from something that is ‘random’.
But it’s highly unpredictable and subject to vast variations.
The system may look disorganized, but there is a set of rules that apply to the chaos.
Deterministic chaos—you may have also heard of it as chaos theory or the butterfly effect.
Weather is actually how chaos theory was first discovered.
Edward Lorenz, a meteorologist at MIT, was performing an early weather simulation in
the 1960s, when he took a shortcut and used input numbers rounded to the nearest thousandth.
He expected to get slightly different results from the version with the full numbers...but
what he got instead was completely different. His investigation into this unexpected outcome
led to the birth of chaos theory, or the idea that chaotic systems magnify even the tiniest
changes in the component parts of that system.
Chaos can make it impossible to accurately predict the behavior of such a system at a
faraway point in time.
You know how you like to complain that the weather report is always wrong?
Blame chaos.
We can never measure the initial conditions precisely enough to accurately predict the
weather past a certain point in time.
That tiny bit off that we will always be is enough to produce a wildly inaccurate prediction
as the system progresses. If sensitivity to initial conditions is one defining factor
of a chaotic system, the other is something called a strange attractor.
Which is just as odd and exciting as it sounds.
Lorenz and his team found that when they ran the weather simulation over and over with
slight variations in input resulting in vastly different results, when you visualize these
patterns, the paths never overlapped.
But at the same time, the paths seemed to circle these empty areas of space—this pattern
is an example of what’s called a strange attractor and one thing that differentiates
a chaotic system from random behavior. With Lorenz’s particular chaotic example, the
visualization starts to look a bit like a butterfly.
The phrase ‘if a butterfly flaps its wings in Brazil it could cause a hurricane in Texas’
is not what the theory is named after, it’s just a memorable example.
This first chaos work required some innovative computing, and although Lorenz typically gets
all the credit for being the ‘father’ of our modern chaos theory, he actually did
this work with two young mathematicians, Margaret Hamilton and Ellen Fetter.
Hamilton later went on to develop the math that got Apollo 11 to the moon, and is credited
with coining the term ‘software engineering’.
So many natural systems are chaotic, like the climate as a whole, the dynamics of clouds,
population dynamics, the patterns of the stock market...the way your milk swirls into and
combines with your coffee?
That’s chaos theory in fluid dynamics. These systems may be chaotic and have seemed impossibly
daunting in the past.
But the math of chaos theory is now small potatoes to the huge supercomputers that we
can use to calculate the progression of chaotic systems like the climate with more accuracy
than ever before.
Even outside of modeling, chaos theory proves exceptionally useful in other fields...like
encryption. And as we move into brave new worlds of exascale computing, quantum computing,
machine learning and other kinds of artificial intelligence, it’ll be exciting to see just
how far we can push chaos theory to help us predict the behavior of chaotic systems...essentially
looking further into the future than we ever have before.
If you want more on unusual math, check out our video on fractals over here, and subscribe
to Seeker for more deep dives into tangled topics like this.
Let us know what math subject you want us to tackle next in the comments below, and
as always, thanks for watching.