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