Mathematical observations

[Experiments,General] (12.11.08, 5:16 pm)

So, I spent some time fiddling around with the Henon applet. It’s possible to discover and get a feel for several properties of the map just through experimenting, and that’s a really positive sign. You can easily identify the two fixed points, and it’s possible to tell what their relative stabilities are. It’s also possible to make some other interesting visual observations.

One thing that stands out in mind is a way in which the Henon map is different from, let’s say, a Julia set. The filled in gray region represents the space where the orbit of every point in the set does not diverge. Generally, points in this region will fall into the characteristic parabola saddle shape shown by the white points. This means that points in this region have a chaotic orbit. Points in this region that are somewhat apart will eventually be separated no matter how close they start together. However, if you choose any point on the border of the gray region, it will converge to the unstable fixed point on the left of the map. Even though this point is unstable along one axis, it is very compressed along the other. I don’t have a proof for this, but it seems visually evident.

What is interesting about this, is that the situation here is exactly the opposite of Julia sets in the canonical z->z^2+c map. In these cases, the border of the filled-in Julia set is chaotic, any two points on this border will eventually become separated. Whereas points within the filled-in set will always converge to some cycle.

It seems, given this, that the filled-in region is actually the Henon map’s Julia set, whereas the border is the “filled-in” Julia set. Or maybe there’s a different term for it. It’s been a while since I’ve done math, so it’s hard to know.

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