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We're gonna start our exploration of the
field of nonlinear dynamics by studying
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maps - systems that operate in discrete
time. And then we'll move to the study of
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flows - continuous time systems.
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This distinction may not be something
you've come across before.
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A flow is something like my pendulum, dynamics
that operate continuously in time and space.
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Imagine, though, if you shone a strobe light
at the pendulum. Every tenth of a second,
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that strobe light would illuminate where the
pendulum was.
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That is a map. Time only exists in that
dynamical system at discrete intervals.
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In other words, it doesn't make sense to
ask what the state of the system is in
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between the samples. Monthly economic
indicators are like this as are old-fashioned
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movies which were shot at twenty-four frames
per second. There was no picture of the state
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of the system in between those frames.
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Here's a slide that summarizes that distinction.
The reason to study maps first, by the way,
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their dynamics are representative. A good
example of what can happen in nonlinear
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dynamical systems, but the math is a lot
easier.
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Most nonlinear dynamics courses take this
path for that reason: introducing the ideas
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and examples in the context of maps and
then circling back around through those
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ideas in the context of flows.
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The map in that previous slide is a mathematical
operator that advances the state one-time click.
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That is, it takes the current state of the system
and tells you what the next state will be.
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Mathematically, we describe this using
what's called a difference equation.
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Here, n is time, x is the state of the system,
f is the map that takes the current state x_n,
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and moves it one time-click forward, giving
you x_n+1.
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Difference equations are very different than
differential equations which we will get to in
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Unit 3 of this course.
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Here's an example of a difference equation.
This is, for obvious reasons, called the
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cosine map, and the way that you would
implement it is with the cosine key on a
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calculator. For a simple code loop, it
takes the cosine repeatedly.
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If, for example, you plugged in forty-eight
degrees to your calculator or the corresponding
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number of radiance, and then you hit the
cosine key three times, what you would see
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if you were using my calculator, is this
value in the display. And then if I hit the
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cosine key a fourth fifth and sixth time
that value would not change.
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Now let's imagine plotting the progressions
of the iterates of this map as a function
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of n (time). This is what you'd see. The
red arrows on the bottom plot are the action
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of the cosine map on the previous point .
Now, your mileage may vary. You may
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get 0.9936957. It depends on how your calculator
or your computer implements the cosine operator.
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We'll get back to that as well. One very
important notion here is that the iterates
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of the map converged to a fixed value,
and then don't change.
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That's called a fixed point of the map. Here's
another difference equation, it's called the
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logistic map. Many of you may have seen this
especially if you took Melanie's course on
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complexity. Here again x is the state of the
system, n is time, and the map has a parameter
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It's called R, and the map will do very different
things for different values of R, as we will
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play with over the next couple of segments.
This map, by the way, is a very very simple
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population model. Again, this is covered in
far more detail in Melanie's course.
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You can think of x as something like the
ratio of foxes to rabbits in my backyard,
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and R is something like the ratio of the
number of rabbits a fox eats per year and
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the number of babies a rabbit has per year.
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That is not a direct correspondence. Again,
you can see Melanie's course for a lot better
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treatment for that model. For the purposes
of this course, this is mostly just an example
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to play with. Now in this equation, x runs
from 0 to 1. Those are all equivalent mathematical
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statements that go with the words I just
said.
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N, again, is time, it is discrete, it is
integer valued, and R can range from 0 to
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4 before the map blows up. The logistic map
has one state variable, so, framed
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mathematically, the logistic map maps the
unit interval to itself, like this.
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We'll get back to what I mean by mapping an
interval to itself a little bit later on.
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In the mean time, let's plug in a few x's
and see what happens.
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Let's say that the first x is 0.2, and let's
say that we're gonna try out R=2.
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Let's see what happens. How I get x_1 is
by plugging in, and what I get
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when I do that is 0.32. To get x_2, I plug
x_1 back into the logistic map, like this.
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And I can keep doing this and something
interesting happens. As we iterate this map,
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the iterates of x_n approach, again, a fixed
point. Let's plot this behavior as we did with
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the cosine map.
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Again, you can see that the behavior has
converged to a fixed point at 0.5, after
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going through what is called a transient
phase.
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Here's an app you can use to explore this.
At the top left, you can see the webpage link.
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That link is also on the quiz that follows
this video, so don't worry about writing it down.
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Now this app has a lot of functionality that
we'll use this unit and next unit.
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This week, what you need to pay attention
to is the right-hand plot, right here.
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This box, this box, and this box, and this button.
This tells you how many points you wanna
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iterate the map. This tells you where you
wanna start, and this tells you what R value to use.
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Say, you enter in, let's see, I think we
started from 0.2 before and we used an R of 2,
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And I think I did five iterates, so I'm gonna
restart this simulation and I'm gonna get
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back a nice computer version of the plot
that I did very badly by hand.
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Let's try plotting a few more points to see
if that fixed point holds still.
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Looks like it does. Let's try a different
initial condition and see if it goes to the
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same fixed point.
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Looks like it does. Let's try changing R a
little bit and see what happens.
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Oops, looks like the fixed point is not the
same place.
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First of all, look at this, fixed point's
at 0.5.
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Fixed point's up a little higher. So the
fixed point moves under the influence of the
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parameter, R. And you can imagine this as
a population that stabilizes as a certain
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ratio of foxes to rabbits in my backyard,
as we change the birth rate of rabbits,
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and the hunger of foxes. It makes sense
that that fixed point would raise and lower
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as I change these parameters.
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Next time we'll explore a little further out
the R x's and see what happens there.