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