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.