Exploring the Volterra-Lotka Fractals
Brian's Map gives and indication of the type of fractal generated by each pair of values for h and p between 0 and 1
The default settings in the Volterra program are h=0.739 and p=0.739. This gives a fractal with a "wiggly line" attractor, a nine-point cyclic attractor and a single unstable point at (1,1).
|h=0.739 p=0.739||h=0.644 p=0.58|
The volterra program only distinguishes between points with a finite attractor (red) and those without (yellow). The two pictures above show the basins of the different attractors, and also the time to convergence.
In the picture (above right) for h=0.644, p=0.58
you will see the basins of three different attractors,
coloured red, blue and purple. The red and blue attractors
are cyclic attractors of periods 42 and 11 respecrtively.
The purple attractor is in the form of an number of disjoint broken arcs.
x plotted from 1.6 to 1.7
y plotted from 2.2 to 2.4
In the picture on the right the purple area is part of the basin for the broken arc attractor.
Try this with your programme :-
Set h = 0.644, p = 0.58 and plot. Now set orbit length to 10,000, cancel show end points, check show last 1000 points and move the dialog box out of the way at the top right. Set orbit start to ( 2.59 , 1.605 ) and draw.
O.K. that gives you a 42 cycle. Reset orbit start to ( 2.99 , 1.875 ) and draw. That gives an 11 cycle round the outside of the 42.
We have had all that
before. Will every internal point go
into one of those cycles?
Of course ... why is he asking!
Cancel the orbit lines and set orbit start to ( 2.244 , 2.514 ) and draw. What are those points? Increase the iteration to 100,000 and show last 10,000 points. You get a few more to join in with the others.
O.K., increase iteration to 1,000,000 and show last 100,000. Not much change noticeable. In fact this collection of points seems to be a limiting collection. You get to the same set from other starts, e.g. ( 1.916 , 1.975 ). I was speculating whether a limiting orbit set could be other than a finite set (period k>=1) or a strange attractor in the form of a continuous closed curve. Here we appear to have a strange attractor in the form of a finite number of dis-joint arcs. Redraw with orbit lines on to see what I mean.
Notice the gap in each arc - this is permanent.
Brian Edwards has created a map of what happens for all combinations of values of p and h between 0 and 1. Click here for Brianís map.