What are the lines on a Bifurcation Diagram?

Solution 1:

these are criticval value curves = iterates of critical value. See also

• wikipedia page about complex quadratic polynomial
• An Introduction to Q-curves in the Bifurcation Diagram by C Ross

Solution 2:

These are called critical curves or Q-curves. They appear so dark because they have a high concentration of x-values. The following histogram shows the frequency distribution of x-values for r = 3.6785:

This matches with the darkness of the bifurcation diagram at r = 3.6785. To explain why there are denser regions in the bifurcation diagram, we will now try to find equations for the critical curves.

The following animation shows the orbit of many random initial values:

As you can see, with each iteration a new critical curves becomes visible. After one iteration we get a straight line Q₁ that also serves as an upper bound for all x-values. Its slope is r/4, which makes perfect sense because the logistic map has its maximum at the critical point x = 0.5. Plugging that into f (x) = rx(1−x) gives r/4.

Why is there a higher concentration of points near Q₁? This becomes obvious when you apply the logistic map to a bunch of points on the unit interval:

The inverted parabola is the least steep around x = 0.5 so this region has the highest density. Since most points get mapped near the maximum value r/4 during the first iteration, we see a dense curve Q₂(r) = f (r/4) after the second iteration. If you plot Q_n(r) = f ⁿ(0.5), where f ⁿ denotes the n-th iteration, you get the familiar looking curves:

The critical curves have some useful properties:^{[1] [2]}

• Two curves cross at a Misiurewicz point (an r-value where the orbit of the critical point is stricly pre-periodic, i.e., it eventually falls onto a periodic orbit but is not periodic itself).
• Two curves touch each other in a periodic window.

References

• H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals, 2004, p. 586
• Histograms for 3.57<r<4
• Animation
• Lutz Lehmann's answer