What gives us the lines within the chaos of the logistic equation?

Question

When you look at the logistic map, there are fairly clear lines even within the chaotic parts, which move around like a really weird polynomial. What creates these lines? I'm assuming they attract more points than the lighter parts, giving it the illusion of being more 'defined', but I'm not sure what exactly this is called or how you could work out the equations of these lines if that is possible.

Answer 1

To cite Oteo, Ros: "Double precision errors in the logistic map", Physical Review E76, 036214 (2007)

The darker lines clearly visible in Fig.8b correspond to peaks of $\rho(x)$. They are known as boundaries [24,25] and are given by the curves $f^{[n]}(1/2,r)$, the locus generated after $n$ iterations of the logistic map starting at its critical point $x=1/2$. For $n=1,2$ they form the exterior boundaries of the bifurcation diagram for $r>r_{\infty}$ and are the lines $r/4$ (upper-most boundary) and $r(4−r)/16$ (lower-most boundary). For $n\ge3$ they are termed internal and their crossings correspond to fixed or periodic points (stable or unstable) eventually attracting the critical point.

_{$\rho(x)$ is the invariant density for a fixed $r$, $r_\infty$ is the limit point for the initial doubling bifurcation sequence.}

Plotting these curves for $n=1,..,7$ gives the convincing picture

Additional iterations make the diagram more chaotic, esp. towards $r=4$ (as expected).

The additional citations are to

• 24: R. V. Jensen and C. R. Myers: "Images of the critical points of nonlinear maps", Physical Review A 32, 1222-4 (1985)
• 25: Eidson, J. ; Flynn, S. ; Holm, C. ; Weeks, D. ; Fox, R. F.: "Elementary explanation of boundary shading in chaotic-attractor plots for the Feigenbaum map and the circle map", Physical Review A 33, 2809 (1986)