bifurcation diagram

The algorithm for drawing the orbit diagram is described in many places, so we'll make it quick here, with the numbers actually used: at each of 4000 equally spaced values of c from left to right in [-2.25, 0.375], the first 2000 members of the orbit of zero for fc(x) = x2 + c were computed, and then the next 200 members actually plotted. It was expected that for c values at which there is an attracting periodic cycle, by 2000 iterations the orbit of zero should have become attracted; otherwise 200 scattered iterates indicate chaotic behavior. Therefore, chaotic "regions" and attracting periodic points appear. It took my program about 10 seconds to make the plot. Some parts of the two pictures are identical: namely at the periodic points which are attracting. The rest of the points in the bifurcation diagram show repelling periodic points. Notice how "dense" they are along the vertical line at c =-2! Note how they fall into "Cantor sets" on any vertical line to the left of -2! These observations are fully discussed in our CMJ article. Note that both curves are criss-crossed left-to-right by implicitly-drawn polynomial curves, called " Q-curves ". They interact in a beautiful way with the bifurcation diagram, and this is the subject of a paper to appear. It's easy to modify the algorithm to make other useful plots. Consider the examples below. Click here to see the whole figure and more on our Superposition Page. The image on the left shows both the bifurcation and orbit diagrams together on the same plot. Periodic points on the bifurcation diagram were colored as in the figure above, so fuschia and blue represent repelling points, and yellow and orange show attracting points. The bifurcation diagram was plotted using thick dots. The orbit diagram was plotted in green. At attracting periodic points, it "agrees" with the bifurcation diagram. This agreement is seen as the thin green lines centered in the yellow and orange curves. Click here to see the big picture of which this is a little piece. Click here to see something you might not expect. The logistic function is often used to model population growth, so negative values aren't usually discussed. But there's a bifurcation diagram and an orbit diagram on each side of the vertical axis. And something else you might not have seen, in addition to a "transcritical" bifurcation. Check it out.