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.