Non-linear recursive functions

[Richard Clark]

I've been investigating orbits produced by iterating funtions of the
form f(x,y) = (y,g(x,y)) for different functions g and different
initial values of x and y.

For example let g(x,y) = 2^y - x

f then has 2 fixed points; at (1,1) and (2,2)

(This is quite easy to do in Excel.)

If we start from the point (1+a,1+a) where 0 < a < 1 the orbit goes
round the point (1,1) in a loop if a is close to 0. As we increase
the size of a the loop seems to get 'pulled' towards the other fixed
point (2,2) so that it has a pear shape. As a gets very close to 1
(e.g. 0.999) an interesting thing happens: The orbit goes round (1,1)
in a loop a certain number of times and then shoots off extremely
quickly. This seems to be chaotic: Although the same behaviour occurs
if we increase a further, the number of times it goes around the loop
before it shoots off is unpredictable.

Does anybody know anything about these functions?

Is there a general theory of them?

[Tim Norfolk]

Look up 'ergodic theory'.

[Richard Clark]

Thanks, but I don't think ergodic theory applies to the function above because Lesbesgue measure isn't invariant under it (although I think it may be quasi-invariant).

[Tim Norfolk]

If this were my problem, I might start with a local linearization of the operator, to determine which of the fixed points were attractive.

[Richard Clark]

>
> If this were my problem, I might start with a local linearization of the operator, to determine which of the fixed points were attractive.

Yes, I've looked into that; In fact neither of the fixed points are attractive. I have got somewhere with the problem. It seems that there is an attractor at infinity whose basin of attraction is the whole of the plane apart from the two fixed points and the points that go around the fixed point (1,1). Its boundary is a closed curve and points on the boundary keep going round on it. Points in the basin close to the boundary then get pulled around it an unpredictable number of times before being allowed to shoot off.

[Richard Clark]

On Monday, July 9, 2012 11:59:47 PM UTC+1, Richard Clark wrote:
> >
>
> > If this were my problem, I might start with a local linearization of the operator, to determine which of the fixed points were attractive.
>
>
>
> Yes, I've looked into that; In fact neither of the fixed points are attractive. I have got somewhere with the problem. It seems that there is an attractor at infinity whose basin of attraction is the whole of the plane apart from the two fixed points and the points that go around the fixed point (1,1). Its boundary is a closed curve and points on the boundary keep going round on it. Points in the basin close to the boundary then get pulled around it an unpredictable number of times before being allowed to shoot off.

I've got further with this.

It turns out that the fixed point at (1,1) is a centre and the fixed point at (2,2) is a saddle point.

If we replace the function g by the function h, where h = 1-x if y<=1 and h = 3y-2-x if y>1 we get a piecewise linear map which has a centre at (0.5,0.5) and a saddle point at (2,2) that displays similar behaviour.

We can analyse what is happening as follows:

1. Take the triangle with vertices at (2,2) and at the points where the eigenvalues of the map that defines the top part the map cuts the line y=1.

2. Analyse what happens to the triangle when we iterate the map.

There doesn't seem to be much stuff about this out there. Does anybody know if there is a classification theorem for discrete dynamical systerms in 2 dimensions in terms of their fixed points?

[Richard Clark]

I meant eigenvectors not eigenvalues

[Tim Golden BandTech.com]

Your equation in terms of complex form has a rotation in it:
( x + iy ) = i( y - xi )
but you've got the 2^y added in there, which I don't know what to do
with in these terms.
Have you done some computer graphical analysis?
I think that could help.
I might have some time to try this out.
Should be pretty stable for negative y, so that a Mandelbrot style
test could yield some graphics.
There are fractal programs out there for things like this. Test
lattice points and see where it escapes.

- Tim

[Richard Clark]

> Your equation in terms of complex form has a rotation in it:
>
> ( x + iy ) = i( y - xi )
>
> but you've got the 2^y added in there, which I don't know what to do
>
> with in these terms.
>
> Have you done some computer graphical analysis?
>
> I think that could help.
>
> I might have some time to try this out.
>
> Should be pretty stable for negative y, so that a Mandelbrot style
>
> test could yield some graphics.
>
> There are fractal programs out there for things like this. Test
>
> lattice points and see where it escapes.
>
>
>
> - Tim

I've investigated it in Excel as follows:

Place initial value of x in cell A1
Place initial value of y in cell A2
Place '=2^A2-A1' in cell A3
Place '=A2' in cell B1

We now drag the formulas in cells A3 and B1 down as far as required and then use a scatter chart to display the two columns.

If we start at (1.99,1.99) the points go round in a loop, apparently forever (I've tried over 100,000 interations) whereas if we start at (1.999,1.999) it
starts going round a loop but then shoots off after about 250 iterations. Between (1.999,1.999) and (2,2) it continues to display the same behaviour, but it's not possible to predict how many times it goes round the loop before it shoots off. Looking at the piecewise-linear map described above helps to demonstrate what's happening, except that the point goes round a square before shooting off instead on round a loop.

Richard

[Tim Golden BandTech.com]

It's a two dimensional system, so you really should test all the
points in the plane, but in computational terms it seems we just need
a lattice of points reaching from say (-3,-3) to (+3,+3).
I've roughed this out just now: there is a tear drop shape of stable
positions in the lower right quadrant.
Points orbit this teardrop shape concentrically, or at least it
appears that they will do this. Because the shape is very simple there
is no need to get too much resolution. So for instance incrementing by
0.1 gives some neat results without a lot of data points; about 3600
test points in this scenario. There is an xy chart available in excel
so you may be able to pull this off. Once you have things setup you
can modify the function easily, but then when you start getting
something fractal or so and you want higher res I suspect you will
want a different platform to do your computations on. There are an
overwhelming number of choices. I found libGd and have been using it
for most of my graphics. I use C++, but that's a tough learning curve.
The recursive stuff is great. Keep going.

I am open to being wrong. It's pretty easy to get a few invisible bugs
going. But I'm pretty sure it's as I describe: like a flower petal a
little ways out from the origin.

- Tim

[Richard Clark]

> It's a two dimensional system, so you really should test all the
>
> points in the plane, but in computational terms it seems we just need
>
> a lattice of points reaching from say (-3,-3) to (+3,+3).
>
> I've roughed this out just now: there is a tear drop shape of stable
>
> positions in the lower right quadrant.
>
> Points orbit this teardrop shape concentrically, or at least it
>
> appears that they will do this. Because the shape is very simple there
>
> is no need to get too much resolution. So for instance incrementing by
>
> 0.1 gives some neat results without a lot of data points; about 3600
>
> test points in this scenario. There is an xy chart available in excel
>
> so you may be able to pull this off. Once you have things setup you
>
> can modify the function easily, but then when you start getting
>
> something fractal or so and you want higher res I suspect you will
>
> want a different platform to do your computations on. There are an
>
> overwhelming number of choices. I found libGd and have been using it
>
> for most of my graphics. I use C++, but that's a tough learning curve.
>
> The recursive stuff is great. Keep going.
>
>
>
> I am open to being wrong. It's pretty easy to get a few invisible bugs
>
> going. But I'm pretty sure it's as I describe: like a flower petal a
>
> little ways out from the origin.
>
>
>
> - 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

[Tim Golden BandTech.com]

I think it's a fine method that you've got. I do think that it is
possible that you could find something new. To me the problem with the
iterated function is in joining them to physics. I enjoy playing with
them, but I'm not sure that we have anything more than pretty
graphics. I do accept the fractal analysis of some natural forms as
related to accumulation, but that does not get us physical
principles.

What is the way into a progression here? It is so easy to get lost in
making graphics, and yet the kernel function comes with no motive. I
don't mean to be discouraging. I'm just wondering, and this criticism
extends onto the Mandelbrot function as well. There must be a bridge
somewhere. Keep going.

- Tim

[1treePetrifiedForestLane]

in deed. just leave the polysignage in hte DumpsterTM;
thank you!

[Richard Clark]

>
> I think it's a fine method that you've got. I do think that it is
>
> possible that you could find something new. To me the problem with the
>
> iterated function is in joining them to physics. I enjoy playing with
>
> them, but I'm not sure that we have anything more than pretty
>
> graphics. I do accept the fractal analysis of some natural forms as
>
> related to accumulation, but that does not get us physical
>
> principles.
>
>
>
> What is the way into a progression here? It is so easy to get lost in
>
> making graphics, and yet the kernel function comes with no motive. I
>
> don't mean to be discouraging. I'm just wondering, and this criticism
>
> extends onto the Mandelbrot function as well. There must be a bridge
>
> somewhere. Keep going.
>
>
>
> - Tim

One way iterated functions are important for physics is in relation to Poincare sections: If a point is moving around an orbit in 3-space, then if we take a plane perpendicular to the motion of this point we can plot the successive positions in the plane that the point cuts it as it moves around the orbit. By looking at the function that describes where these cuts occur we can find out about the stability of the orbit. This can be extended to a point moving around an n-dimensional phase space, in which case the Poincare section will have dimension n-1.

[Tim Golden BandTech.com]

I did try to understand the multidimensional interpretation in quantum
physics back in time, but it didn't stick. I do have math that has
general dimensional properties, and a projection algorithm that can do
n-1 dimensional projection, and back in time I attempted to find
connections to quantum mechanics, but didn't break through.

- Tim

[Richard Clark]

>
> I did try to understand the multidimensional interpretation in quantum
>
> physics back in time, but it didn't stick. I do have math that has
>
> general dimensional properties, and a projection algorithm that can do
>
> n-1 dimensional projection, and back in time I attempted to find
>
> connections to quantum mechanics, but didn't break through.
>
>
>
> - Tim

On a larger scale, Michel Henen used a Poincare map to study the motion of stars in a galaxy.

Richard