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