I want to say a little bit more about different types of fixed points.
For the squaring function, we saw two types of fixed points: stable and unstable.
Here's an illustration of a stable fixed point...
...and here's an illustration of an unstable fixed point.
The fixed point is the black dot, it doesn't change when the function acts on it.
And this shows us that nearby points get closer to the fixed point, hence it's stable.
Here we see that nearby points, when the function acts on them, get further away.
So first, there's some other words for these types of fixed points.
A stable fixed point is also called an "attractor".
It's called an attractor, because it attracts points; nearby points get pulled towards it; you might also say it's an attractive fixed point.
An unstable point is called a "repellor" (or "repeller"), because it repells nearby points, it pushes them further away.
We would also say that this type of a fixed point is repulsive.
There's another way, somewhat metaphorically, of illustrating these different types of fixed points.
So, a stable fixed point, you can think of it sort of like having a marble or a ball resting in the bottom of a bowl.
So, if I took this ball and pushed it up the side a little bit and let it go, it would roll back and forth and come back to this point.
A rock in a valley, a marble in the bottom of a bowl, is a stable situation: a small change, a small push won't change the situation in the long run.
An unstable fixed point can be illustrated with the opposite picture.
Think of a ball carefully balanced on the top of a hill, or a marble on the top of an upside-down ball.
It can be balanced here, it can be fixed, not move, but a tiny little push either to the left or to the right and it will roll down in one direction, and it won't return here.
So, that's the same thing that's showing here; this point is fixed, but if we move a little bit to the right or a little bit to the left, we'll get pushed away, not to return.
So this picture is for a stable fixed point and this for an unstable fixed point.
The distinction between stable and unstable fixed points is important.
In real systems, one would not expect to encounter unstable fixed points, for the simple reason that they don't stick around for long.
A simple bump, a small bump to the left or to the right and the object moves off not to return.
So in real systems and in most numerical experiments, it's stable behavior that we'll observe.
So when we 're trying to categorize and understand the long term behavior of dynamical systems,
we'll be paying special attention to behavior that's stable, since that's what's most likely to observe.