Date: Jan 30, 2013 4:00 PM
Author: Richard Clark
Subject: Re: Non-linear recursive functions
> It's a two dimensional system, so you really should test all the

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> points in the plane, but in computational terms it seems we just need

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> a lattice of points reaching from say (-3,-3) to (+3,+3).

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> I've roughed this out just now: there is a tear drop shape of stable

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> positions in the lower right quadrant.

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> Points orbit this teardrop shape concentrically, or at least it

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> appears that they will do this. Because the shape is very simple there

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> is no need to get too much resolution. So for instance incrementing by

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> 0.1 gives some neat results without a lot of data points; about 3600

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> test points in this scenario. There is an xy chart available in excel

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> so you may be able to pull this off. Once you have things setup you

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> can modify the function easily, but then when you start getting

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> something fractal or so and you want higher res I suspect you will

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> want a different platform to do your computations on. There are an

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> overwhelming number of choices. I found libGd and have been using it

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> for most of my graphics. I use C++, but that's a tough learning curve.

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> The recursive stuff is great. Keep going.

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>

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> I am open to being wrong. It's pretty easy to get a few invisible bugs

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> going. But I'm pretty sure it's as I describe: like a flower petal a

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> little ways out from the origin.

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>

>

> - Tim

The teardrop shape is centred at the stationary point (1,1) and points to the the stationary point at (2,2). Points move concentrically around this teardrop shape (with it becoming more pronounced the further we move out) until we get to about (1.999,1.999),when it goes round the teardrop 6 times and then shoots off to the left. Between (1.999,1.999)and (2,2)similar behaviour is displayed, but we can't predict how many times the point will go round before shooting off.

Points coming in from the top right go straight across, dipping to (2.2) as they pass. If we start low enough they go in a horseshoe below the teardrop.

It's much easier to do in Excel than in a language like C++ because we can put the formulas in and then just drag them down, instead of setting up a loop

and plotting the results.

Richard