Here's another illustration of the butterfly effect
this time using differential equations instead of iterated functions
so we'll go back to the example from the last unit
this example will be a counter example of a non-butterfly example.
Newton's Law of Cooling describes a temperature of a glass of beer
or a glass of anything as it warms up to room temperature.
This is the equation, hopefully by now familiar,
20 is the room temperature, T is the temperature of the beverage.
Reminder, this is a differential equation, rather than changing in steps or clicks
the temperature changes continually with time.
So a solution to this equation,
all solutions will be from Euler's method, or something similar on a computer
The solution to this equation is T(t),
the temperature of the beer as a function of time.
As we have seen, we start with a temperature of 5
we see that the beer warms up, starts at 5, warms
quickly at first it's increasing less quickly and it approaches
equilibrium value, which is stable or an attractor at 20 degrees.
So as we did with the logistic equation
we can think about this a little bit differently.
We can think about using the equation
to make predictions. So if I know
the temperature of the beer now,
I can use this equation to make a prediction
for the temperature of the beer at a later time.
And we might be slightly wrong in our initial temperature measurement
but that difference won’t matter by much.
So let me show that.
Here is a made up example, we have two curves,
one I'm calling Prediction and the other I'm calling Reality.
Doesn’t matter which is which,
but Reality is the curve on the bottom.
Maybe I should color that in in red.
These slides are available online and
they’re in nice color there.
I only had a black and white printer, hence,
the low-tech coloring.
So red Reality and black Prediction and maybe we make an error,
our thermometer isn’t very good,
we somehow overestimate. We think
the initial temperature is 2 instead of 5,
well it’s really not such a big deal.
The temperature difference between Prediction and Reality
starts not so big and actually gets smaller ,
and we know that even if I mismeasure the temperature
we know it will end up at 20°.
If I need to be more accurate say I needed
to know when the temperature got to be 15 for some reason,
well I could measure more accurately.
And maybe I’d need to modify the
differential equation a little bit
to take into account, I don’t know,
that maybe water condenses around the beer as it
starts to warm and that changes the rate of cooling.
So I might need to modify the equation a little bit,
I might need to be more accurate in my initial conditions,
but I’m getting closer and closer to Reality.
Closer and closer to Laplace’s Demon,
so that at least as beer is warming up,
the future can be like the present. We can predict the future with knowledge of the present.
Okay this is the usual scientific gambit
we come up with an equation,
make some measurements,
we can make predictions,
the predictions aren’t perfect but they’re good.
We make the predictions better by tuning our equations,
and measuring more carefully.
Ok, so let’s do another example and this is a famous example
from the history of dynamical systems
and chaos called the Lorenz equations.
These are also differential equations
this time three equations for the price of one,
an equation for x, an equation for y, and an equation for z.
x, y, and z all depend on each other.
So the rate of change of z depends on x, y, and z.
The rate of change of x depends on y and x.
Now mathematically these are a little different
than the differential equations we studied in the last unit
and I will talk more about those in subsequent units
when we talk about strange attractors.
But for now we can think of these as just being
a 3-dimensional or a triple version of the differential equations we have been studying so far.
The origins of these, which I won’t get into,
but as an historical note, they were put forth by Ed Lorenz in the 60’s, MIT meteorologist,
as a very, very simple model for atmospheric convection.
Not really a realistic model,
just like the logistic equation isn’t really a realistic model of population growth
but a very simple model designed to capture some features of convection.
Just like before we can solve these equations, here,
we’ll have to use a computer to do it using Euler’s method
or something a little bit fancier and when I do that
for this equation, I get the following three functions.
So this is an equation for three functions: x, y, and z.
They are each a smooth, continuous function of time.
This is time on this axis and then x, y and z.
The behavior is aperiodic. Just like the behavior
of the Logistic Equation when r = 4 is aperiodic.
So there is some sort of regularity to it but
if you look carefully you'll see that these wiggles don’t quite repeat.
The x and y wiggles are similar,
z is doing something a little differently and looks like
somebody scribbled, got really mad and scribbled like this with a pen.
There is definitely some cyclic behavior but it is not exact.
So these are the solutions, they are aperiodic.
Again, we will talk more about this in subsequent units.
The main point now is to have another picture
to go along with our story of the butterfly effect.
So let’s do the same thing we did before
with Newton’s law of cooling. Let’s think about
using Lorenz’ equations to do predictions. For simplicity
I’ll just focus on the x solution, so we'll just be looking at this curve, not the y and z curves.
Here’s a picture that shows how we might use
the Lorenz equations to think about making predictions.
I’ve plotted here a solution, just one of the solutions,
x, not y and z for two slightly different
initial conditions, 15.1 and 15.0, and actually this should be a minus here,
there’s a typo in this and the next couple of slides,
but the details don’t matter.
So, the Prediction will be the blue curve
and Reality I'll draw in this red, purple, maroon.
The online versions have nice color, but again
I didn’t have a color printer today.
The two curves, Reality and Prediction, are right on top of each other.
There’s not much difference between these
and the two curves agree right around until t=5.
At t=5, the red and blue curves turn apart.
The red curve Reality diverges from the prediction right here
and makes a really sharp turn and then the two curves continue on and they are more or less uncorrelated.
They are wiggling, each doing their own thing
and they seem completely unrelated to each other.
The small differences in initial conditions make a
pretty big difference after 5 minutes or whatever these time units are.
Suppose that we made a better measurement.
Our model is correct, the reality and the model are both
doing the Lorenz equations, following that rule, but the problem
is that our initial condition is wrong. It’s 15.1 when in
reality it’s 15.0. So let’s try a much better initial condition.
So we’ll go from 15.1 to 15.0001 and reality is still 15.0.
Now the blue and maroon, or red, curves are right on
top of each other until around 11.
So Reality and Prediction match each other indistinguishably
for 11 minutes or 11 time steps, and then again Reality takes
a sharp turn away from Prediction and then for the rest of the run
we have very different behaviors of the two curves.
So our prediction is good for a while and then becomes
essentially worthless, rather suddenly, at t=11.
However, we were able to predict farther,
so we improved our measurement, and we can predict now to t=11, before it was just t=5.
So let’s try a really, really good measurement.
Suppose that we use this, 15.00000001
when in reality it is 15.0. So this is
amazingly close to the true value and sure enough,
Reality and Prediction, blue is still Prediction,
red is still Reality, march together right along for further now, we can go until almost 18,
again as before, Reality, the red curve diverges
from the blue curve and then they become very different after this.
So we have a very good measurement
and we can extend our prediction out to 18,
but after 18 our prediction becomes more or less worthless.
I want to stop for a second and
think about the magnitudes of these numbers.
This will help illustrate just what's going on with the butterfly effect.
So let’s say x was measuring length.
Whatever x is measuring the same story will hold
but it will be most concrete if we talk about length.
So 15 meters, that’s really tall,
that’s about as tall as a 5-story building.
But what about this 0.00000001?
That’s about 10 nanometers, 10 x 10-9m.
That’s really small, that’s about a 1,000 times smaller
than a single red blood cell,
10 times larger than a single glucose molecule.
So this says if we have something 15 meters
15 meters + 10 glucose molecules that the behavior
will be completely different after just 18 time steps.
So on the one hand, this is a deterministic system,
just like the differential equations we studied in the last unit,
but it depends so much on the initial condition
that that determinism almost becomes meaningless.
I would say that as a practical matter,
the difference between 15.0 and 15.00000001 isn’t just
one of not having good enough measuring instruments,
it’s almost hard to think physically, what the difference,
how to think differently about what these numbers would mean.
In a physical or practical sense these have to mean the same thing.
Five-story building expands or contracts
much, much, much, more than this every minute or certainly every day.
All sorts of quantities fluctuate naturally,
so if the future depends this much on the initial condition
it’s a different sort of determinism perhaps than LaPlace imagined.
To summarize this portion of the lecture,
this is an example of the butterfly effect.
A very, very small error in the initial
condition grows extremely rapidly. So that means
long term prediction is impossible.
The butterfly effect will eventually win.
All small errors are magnified they are going to grow.
What we have here is a deterministic, i.e., a rule-based system,
that’s behaving unpredictably.
Once we get out past a region of predictability,
even though we have a deterministic system,
it behaves as if it is unpredictable.
a sense where does that unpredictability come from?
Well, this is perhaps oversimplifying a little bit,
but the predictability of the rule,
the rule in a sense is so deterministic,
it depends so much on initial condition
that it requires impossible accuracy,
almost meaningless accuracy in that initial
condition to be able to do prediction.
The predictability or the determinism or the
sensitive dependence on initial conditions is what causes
unpredictability in systems like this that show the butterfly effect.