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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)

distortion

Is there any way to explain this similarity?

Did
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tly
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  • Very nice! Have you read the book Chaos: Making new Science by James Gleick? It covers dynamical systems, including Henon attractor and logistic map. I remember it explained a lot, maybe you would find your answer there and, while looking for it, you would enjoy reading through the rest. – David Apr 24 '13 at 11:40
  • Thank you for the recommendation ;) – tly Apr 24 '13 at 12:54

2 Answers2

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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]"

Foo Barrigno
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Henon gives the canonical definition for the map named in his honor in a 1976 publication in Communications in mathematical physics: $$x_{n+1} = y_n +1 - a*x_n^2 \\ y_{n+1} = b*x_n$$

Substituting, we get $x_{n+1} = b*x_{n-1} + 1 - a*x_n^2$. If $b = 0$, we have $x_{n+1} = 1 - a*x_n^2$, which is just a quadratic map, of which the best known case is the logistic map. The Feigenbaum stuff is important and general, but in this case, the logistic map is simply embedded within the action of the Henon map.

user48117
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