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The Eternal Question

How do you determine how chaotic something is, or in other words, how predictable it is, at any one time?

Introducing the Equation

Where:

> View the further information on this equation, and it's bifurcation diagram.

The Basic Idea

Use May's equation as a source of data ranging from steady-state to chaotic. Take two (close) values for R and calculate them for a certain number of iterations and compare the results.

Simplistic, I know, but it led to interesting results. I still have no idea if they are relevant or accurate, but that's what this page is here for!

In all cases below:

1. There are an initial 100 hidden iterations in addition to that stated.
2. Initial population level Y is set at 0.1

Method 1

Input an initial value for R, and a value for the change in R, called dR.

Iterate 500,000 times. Each iteration calculates 2 new population values:

Y1(t+1) = R1 * Y1(t) * (1 - Y1(t))

and

Y2(t+1) = (R1+ dR) * Y2(t) * (1 - Y2(t))

Y1 is then plotted to the X axis, and Y2 to the Y axis. Each time a point falls on the same spot, the colour of the point is incremented.

PROGRAM: RMAP3.EXE

Results for Method 1

Note that for these images, the origin is in the top left corner.

If dR = 0, the results of the equations will be equal, and you will get a diagonal straight line of points from top left to bottom right.

In the following examples of output dR = 0.001, and you can see the progression from a period two state (indicated by arrows) into chaos:

Summary

First thoughts

It seems to me that this method provides a graphical way to determine how chaotic this system is. The brighter areas of the images above indicate the location of the most popular points during the iteration - which should give you a measure of the probability of a outcome.

In the bifurcation diagram below, you can see that each of the above images has a strong relationship to it:

If you look at the image of R = 3.7, it seems that the horizontal and vertical lines are spaced in a very similarly way to the branching of bifurcations in the image above. What is interesting though, is that the edges of the bands are 'blured', and corners the are 'rounded'. This might imply a tendancy for orbits to move towards one line on another, ie: the lines could be thoughth of as attractors. If that is the case, you would be able to calculate, with some accuracy, how 'chaotic' one value of Y would be.