Here is the Bifurcation Diagram for the logistic map $x_{n+1}=rx_{n}(1-x_{n})$:

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And here is an enlargement of a particular section

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What are the faded lines that appear in the dark sections of the diagram, and why are they so pronounced?

  • $\begingroup$ are you referring to the white strips? $\endgroup$
    – MattW
    Aug 11, 2017 at 15:28
  • $\begingroup$ @MattWatkins No, those are periodic windows. I mean the black curves. $\endgroup$
    – Plato
    Aug 11, 2017 at 18:22

3 Answers 3


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


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:

Histogram with peaks at the boundaries and x=0.728

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:

Mapping the unit interval to an inverted parabola. The points are denser at the parabola's maximum.

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 Qn(r) = fn(0.5), where fn denotes the n-th iteration, you get the familiar looking curves:

The first 7 critical 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.



I think they may be the set of bi-stable, tri-stable, ... n-stable... set of values with a given $r$. Thus these lines are the asymptotic set of values which $x_n$ jumps into, and are condensed with so many points that they become dark. I suppose the set of $x$'s are take with various initial values, but without fully specifying the pseudo code, I don't know how to reproduce the result, thus leaving my quick thoughts as answer to make it more visible only.


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