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