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In the last segment, you saw that the progression
of iterates of the logistic map converged to an
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asymptote. In this segment, I'm going to be a
bit more careful about the definitions and
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terminology around all of that. And I'm
going to show you what happens for different
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values of the initial conditions x_0 and the
parameter, R.
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First of all, that notion of a progression
of iterates, x_0, x_1 and so on.
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That's called an orbit or a trajectory of
the dynamical system.
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An orbit or a trajectory is a sequence of
values of the state variables of the system.
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The logistic map has one state variable, x. Other
systems may have more than one state variable
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My pendulum, for instance, the one you saw
in the very first segment. You need to know
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the position and velocity of both bobs of
the pendulum in order to say what state it's in.
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I'll come back to that in the third unit of
this course.
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The starting value of the state variable in the
logistic map, x_0, is called the initial condition.
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The trajectory of the logistic map from the
initial condition, x=0.2 with R=2, reaches
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what's called a fixed point. That's the asymptote
after going through what's called a transient.
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I drew that picture for you last time.
Here's that picture again.
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Technically, a fixed point is a state of the
system that doesn't move under the influence
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of the dynamics. That is, the fixed point to
which the logistic map orbit converges, is
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what's called an attracting fixed point.
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There are other kinds of fixed points as
I'll show you with my pendulum.
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So this is certainly a fixed point of the
dynamics. The system is there and the
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dynamics are not causing it to move. And it's
an attracting fixed point, because if I perturb
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it a little bit, that perturbation will shrink,
returning the device to the fixed point.
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Now, that's an attracting fixed point. As I
said, there are other kinds of fixed points.
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This is one of them. Or, there is one here.
I've never gotten the pendulum to sit at it.
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There is some point here for the pendulum
where it will balance.
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So that is a fixed point in the sense that
the system will not move from there,
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but it is an unstable fixed point.
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There are two other unstable fixed points
in this system. This one, and this one.
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Again, all of these points are states of the
system that the dynamics is stationary.
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This definition that I just gave you captures
both kinds of fixed points.
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States that don't move under the influence
of the dynamics, but doesn't tell you whether
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they are stable, that is, they are attracting,
or they are unstable, that is, they are repelling,
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like the inverted point of the pendulum.
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Dynamical systems have several different kinds
of asymptotic behaviors.
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Subsets of the set of possible states to which
things converge as time goes to infinity.
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These are called attractors.
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Attractors, by the way, have a somewhat
circular definition as what's left after the
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transient dies out. There's a way to formalize
that, which I can put up on our auxiliary video,
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if people are interested.
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Attracting fixed points are one kind of attractor.
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There are three other kinds. We'll talk about
some of those in the next segment, and all
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of them over the course of the next two weeks.
Now, back to fixed points.
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Remember this demonstration? Using the logistic
map application, that showed that lots of
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different initial conditions go to the same
fixed point.
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So if we use the initial condition 0.1, and
the parameter value 2.2, we go to this
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fixed point. Let's try something different.
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Different transient, same fixed point.
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Different transient, still goes to the same
fixed point.
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The way we think about that behavior, a whole
bunch of initial conditions going to the same
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attractor, is by defining something called
a basin of attraction.
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If you are from the United States, there's an
easy analogy for you to understand this.
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In the middle of the United States, there's
something called the continental divide.
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It runs about ten miles west of where I am
sitting right now, and a raindrop that falls
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to the west of the continental divide will
run down to the Pacific Ocean.
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A raindrop that falls to the east of the
continental divide will run down to the Atlantic
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Ocean, or maybe down Mississippi. and out
that way.
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The analogy here is that the Atlantic Ocean
as an attractor and the terrain to the east
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of the continental divide is the basin of attraction
of that attractor.
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The Pacific Ocean is another attractor, and
the terrain to the west of the continental
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divide is the basin of attraction of that
attractor, and the boundary of the basin
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of attraction divides those two basins.
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What do you think will happen to a raindrop that
falls exactly perfectly on that basin boundary?
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Now let's go back and explore what happens
if we change the R parameter while keeping
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x_0 fixed, that is, using the same initial
condition. There's R=2.3, R=2.4, R=2.5,
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as I mentioned in the last segment, the
fixed point moves. That's like the population
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of rabbits stabilizing at a higher number
if the foxes are less hungry or the rabbit's
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birth rate is higher.
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Now if you look closely, you'll see that the
transient lengths differed in that experiment
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I just did. R=2.2, the population stabilized
really quickly. It took a little longer at R=2.3.
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The analogy there is that the population
takes a little bit longer to converge to its
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fixed point ratio of foxes and rabbits. You
also may have noticed, this little overshoot
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right here, which gets more pronounced if we raise
R further.
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There's R=2.6, R=2.7, what's going on here
is that the orbit is still converging to a fixed
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point, but instead of converging in a one-
sided fashion, it's converging in an oscillatory
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fashion. It's kind of like, if you push down
on the hood of your car, and the car bounces
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up and down for a while, before settling out.