Similarity between Henon attractor and logistic map?

Question

I noticed that layering Henon attractor images with avalue=1 and bvalues from -0.2 to 0.3 looks like a distorted version of the logistic map. In the image below you can see the layered images (left). After that I tried to accentuate the first "branches" (right)

Is there any way to explain this similarity?

Answer 1

There is a way to explain the similarity. According to wikipedia, the Feigenbaum constant is tied to the bifurcation of all chaotic maps:

http://en.wikipedia.org/wiki/Feigenbaum_constants

"Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate. It was discovered in 1978.[1]"