Hello and welcome to unit 3 of the course.
This unit is on chaos and the butterfly effect.
When I teach these topics at College of the Atlantic,
I like to keep this unit a bit of a surprise
so students don't know the butterfly effect is about to hit them.
I can't do that here because I had to give these units a name ahead of time
and then the name is printed on the screen for you to see.
So it won't be a surprise, but it'll still be awesome.
There's all sorts of interesting stuff and big ideas in this unit.
I'll begin the course by introducing two types of dynamical systems.
Remember a dynamical system is just a rule for how something changes in time,
the two types of dynamical systems,
or iterative functions and differential equations.
The main example I'll used to illustrate the butterfly effect in this unit
is an iterated function called the logistic equation.
So, I'll begin this unit
by presenting the logistic equation
and introducing it in a number of different ways.
Then we'll encounter chaos in the butterfly effect,
then I'll be in a position to define those terms fairly carefully.
Next there'll be an optional lecture on Lyapunov exponents,
and then we'll conclude by thinking about the implications of the butterfly effect
and digging deeper into the very idea of randomness itself.
So let's get this unit started by presenting the logistic equation.
The logistic equation is a simple model of population growth.
It's an iterated function,
and it tells us how the population might change from year to year.
So, the picture is is that the population next year
is a function of the population this year.
And we might write that function as f(P).
So P will be population,
and f(P) tells us how we get the next year's population
given this year's population.
Again this is an iterated function, so time is discrete:
we're not monitoring the population at every instant as it flows up and down,
but instead we're just measuring it once a year
or once a generation or something.
So before I present the logistic equation, I want to present an even simpler model
that leads to exponential population growth.
So suppose we have a situation where a population doubles every year.
So the population doubles every year and f(P) is 2P.
The population next year is twice the population this year.
I can write that in another notation that perhaps makes this clear.
P_{n+1}, that's next year's population, is just twice of this year's population.
So this is just the doubling function,
one of our first examples of an iterated function from unit 1.
So just as a reminder of how this goes, and to make this a little more concrete,
let's say we're talking about a population of rabbits on an island.
So, on an island, maybe initially,
somebody leaves some rabbits there by mistake,
some rabbits escape, we would have two rabbits.
Then we'll come back the next year to this island,
and we would have 4 rabbits.
Next year those 4 rabbits double again, and we would have 8 rabbits.
And you can see the rabbits are taking over the island,
they're taking over the page in this case.
So the rabbits are growing and growing and growing,
they're doubling at every time.
So let's just do that with numbers.
Get rid of the rabbits.
So if we had, we chose, an initial population, a seed,
then that would be ... we chose that to be 2,
and then our next value we double 2 to get 4,
and our next value we double 4, 4 times 2 is 8,
I can keep going ...
and the rabbits ...
So we keep doubling, the rabbits will grow without bound,
we would say that P_n tends towards infinity.
Eventually the world would be nothing but rabbits.
We can generalize the simple model of population growth as follows:
In the previous model we had the population doubling every year,
I could be a little more general and say f(P) equals rP.
So rather than multiply the population by 2 every time,
I could multiply it by some other number r.
So the picture is that the population next year
is this function of the population now.
Or P_{n+1}, next year's population, is just r times this year's population.
So again r is the growth rate.
So there are three cases that we might be interested in,
three different behaviors depending on the value of r.
So if r is greater than one, as was the case previously,
the population will grow continually and tend towards infinity.
So r larger than 1,
that means every year the population increases,
this year's population is larger than last year's,
next year's will be larger still.
In that case the population tends towards infinity,
it grows without bound.
On the other hand if r equals 1, the population stays the same.
So if r is 1, we just multiply the population by 1,
that doesn't change the population at all.
So then for this value of r, any population would be fixed,
it doesn't change.
And then if r is less than 1, but greater than 0,
then the population approaches 0.
So if r is, say, 0.5,
that means next year they'll be half as many rabbits as there are this year.
The following year they'll be half as many again, and so on,
so the population is getting smaller and smaller and approaching 0.
It's bad news for the rabbits:
the rabbits are going to die out on the island.
And in this model, since we're thinking about populations,
we'll keep r and P always positive.
So we won't worry about negative P or negative r.
So this is a very simple model of population growth,
that's not very realistic, but it's a starting point,
and note that it has three very different behaviors
depending on the three different r values.
And in this context I should mention
that the quantity r is sometimes called a parameter.
I guess I'll write that here.
So a parameter in a model is something that one might change.
You might change it to explore different behaviors in the model, as I've done here.
We see it does different things for different values of r.
Or if you're trying to model a real situation,
you might adjust the parameter r until you best fit the data.
In any event r is a parameter for this simple model.