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To illustrate the idea of stability, lets do another example.
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The function g of x equals half x minus 4 has a fixed point at -8.
3
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And you can check this if you plug in -8 to this function, you 'll get -8 out.
4
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So it has a fixed point at -8. How could we determine its stability?
5
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So, a fixed point is stable, if nearby points get pulled towards it,
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and it's unstable if points get pushed away.
7
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So lets try doing an experiment with an initial condition near - but not exactly at - this fixed point and see what happens.
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I would choose the seed of -9.
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What's my next value? Well, the function tells me that: I take -9...
10
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...-8.5.
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What's the next value? Well, I just do the function of that. Iteration is doing the same thing over and over again: take the value, divide it by 2, subtract 4...
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... -8.25. It looks like it's getting closer to -8.
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Lets do one more. What's f...or g, excuse me, of -8.25? Divide by 2, subtract 4...
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...and, as you might have guessed...
15
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So, the seed, the initial condition -9 is getting closer to -8.
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So that leads me to believe that is stable. Just to be sure, lets check an initial condition on the other side of the fixed point.
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What happens if we iterate -7? Well, lets see. I can just do it again on a calculator.
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We see that -7 also gets closer to -8. So, orbits or initial conditions either side on this fixed point, -8, get closer to it.
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I can make a really rough sketch of that on a time series plot.
20
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There are my axes. The point -7 gets closer to 8,
21
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and the point -9 also gets closer - I should say - to -8.
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If I connect the dots, I 'd see something like this. Both of these initial conditions, -7 and -9,
23
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there's -7, -9, get closer to -8 as time gows forward. And I could also draw a phase line for this.
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Here's my -8, that's my fixed point, and nearby points get pulled towards it.
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So we would say that -8 is a stable fixed point - stable because the arrows are getting in, orbits are getting pulled towards it.
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It turns out for this function that there are no other fixed points, so this is the entire story:
27
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any initial condition will get pulled in to -8.