The Volterra Program
I have written a little program to explore the fractals generated using different values of p and h. In the picture above the default values (p-0.739, h=0.739) have been used. For each point on the plot the values of x and y at that point are fed into the formula for the iterative method. For some points (coloured yellow) the method is unstable and the new values for x and y get larger and larger. For other points, the values of x and y remain small, and these points are coloured red. For any point in the red area, the successive points calculated by the iterative method will also all fall in the red area. I call this sequence of points an orbit, and in the illustration above you can see the points of one such calculation, coloured lime green and joined by black lines.
The orbits of all the red points must lie in the red area, and in particular, the final point of the orbit, after the maximum number of iterations (the default is 100) will also be in the red area. I have coloured all such final points in blue. These blue points can be divided into two sets. Most prominently there is a wiggly line in the form of a closed loop. Round the outside of the wiggly line are nine separate points. These nine points are more difficult to see, so I have drawn the orbit for one of them. This orbit just goes round and round the nine points. An orbit started at almost any other point in the red area gets sucked into the wiggly line.
I find this amazing enough. But Brian has gone on to discover more curiosities. Visit the exploration page to find out.
And if you would like to try the Volterra program yourself, you can download it.volterra.exe is a Windows95/98 Program.
Last updated February 04 2001