There are two numbers that don't change when they're squared: 0 and 1.
1 squared is 1 times 1, and that's 1.
So, you could say f of 1 is 1.
And 0 squared, that's 0 times 0, which is definitely 0
So we could say that f of 0 is 0.
So these are fixed points, points that do not change when iterated.
So a fixed point of a function f is a number that does not change when iterated.
The squaring function has two fixed points, 0 and 1
because 0 squared is 0, input equals the output
and and 1 squared is 1, input equals the output.
So now we're ready to state the dynamics,
the long term behaviour of all orbits all at once.
And we'll do that with a geometrical construction called a phase line.
So here's the phase line for f of x equals x squared.
In this analysis I'm only going to worry about positive x values.
So I'm going to draw my line, and this will be a number line.
Here, the point 0, I'll draw that as a dot because 0 is a fixed point
And I'll draw 1 as a dot because 1 is a fixed point.
Any number larger than 1
gets larger, moves to the right on this number line when it's iterated.
And if we start with a number between 0 and 1 it'll get smaller.
So this is the phase line for the squaring function.
It tells us the long term behaviour, the ultimate fate, of any initial condition,
or any positive initial condition for this function.
If I start anywhere greater than 1 I'll move to the right
for ever, I get larger and larger.
If I start anywhere in here, between 0 and 1, I'll move to the left and get closer and closer to 0.
We say that orbits in here approach 0;
orbits here tend towards infinity or grow without bound or diverge
The point 1 is a fixed point, it stays put, that's why it's a dot
and the point 0 is a fixed point.
So compare the phase line to what I drew here
with the time series plots.
They contain more or less the same information.
Here on the time series plot
we saw that these two initial conditions,
and we could plot additional points,
any initial condition up here, get's larger, moves up.
And any initial condition in here
gets smaller and approaches 0.
So these two plots show similar things
but in somewhat different ways.
One thing to note is that on the timeseries plot
we can see how quickly this gets large.
It's getting large pretty quickly in fact,
this point would be off the screen entirely.
On the phase line we just know the direction of motion.
We don't really know the speed.
All we know is that orbits move this way
and that orbits move this way.
The phase line tells us the direction of motion
but not the speed.
So phase lines and their higher dimensional
analogues, are very useful geometrical constructions
for describing the dynamics of a function.
This completely describes the long term dynamics
of the squaring function for positive numbers
So to summarise one more time
The phase line tells us the following things:
seeds larger than 1 tend towards infinity
seeds between 0 and 1 approach 0
1 and 0 are fixed points
but they are different types of fixed points.
1 is and unstable fixed point.
Unstable because if you're at 1 and you move
a little bit to the left or to the right
you don't return, you get pushed away.
So that's an unstable situation,
it's like being perched on the top of a hill -
a little push and you'll roll down in either direction.
In contrast, we say that 0 is a stable fixed point.
Stable because if you're at 0 and you move
a little bit to the right, you get pushed right back to 0.
so that's a stable fixed point - it's a stable situation.