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A lot of natural phenomena can be modeled with Logistic Map or a similar map as they universally show transition to chaos. Logistic Map maps bounded [0,1] -> [0,1] intervals. Is there an analog with the same CHAOS typical behavior but for unbounded [-∞,∞] -> [-∞,∞] intervals? I derived a few but they seem too complex. Anyone knows some simpler cases? The reason I am asking is that in reality unbounded intervals / domains / values happen more naturally. I know I can map/rescale infinite to bound values, but I want to see what is the simplest analytic formula for unbound domains reproducing Logistic Map universal "transition to chaos" behavior.

Example of derivation: Tanh

Basic idea is to use Logistic Map formula but to map between [0,1] and [-∞,∞] before and at the end of iteration. This is a kink mapping [-∞,∞]->[0,1]:

$$\frac{\tanh (x)}{2}+\frac{1}{2}$$

Substitute to logistic map and simplify (Wolfram Language code):

a z (1 - z) /. z -> 1/2 + Tanh[x]/2 // FullSimplify

$$\frac{1}{4} a \text{sech}^2(x)$$

Now substitute back into the inverse function taking back [0,1]->[-∞,∞]:

ArcTanh[2 z - 1] /. z -> 1/4 a Sech[x]^2 // FullSimplify

$$\tanh ^{-1}\left(\frac{1}{2} \left(a \text{sech}^2(x)-2\right)\right)$$

Here is how it looks and it does seem to work properly. But this looks a bit convoluted. Any simpler analytic formulas for unbounded Logistic Map showing same CHAOS behavior, for instance leading to Feigenbaum Constant? Logistic Map behavior is universal and will be found in any similar map. So I guess modifications of Logistic Map or bound<->unbound mapping are OK if we recover bifurcation diagrams, chaos, Feigenbaum Constant, etc, on unbound domains.

enter image description here

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  • $\begingroup$ Hmm, would $x_{n+1}=\lambda \sin(\pi x_n)$ count? $\endgroup$ – Anders Sandberg May 14 at 10:44
  • $\begingroup$ @AndersSandberg thanks for the idea, but it maps [-∞,∞]->[-𝜆,𝜆] and I am looking for [-∞,∞]->[-∞,∞] with a simple formula where constant parameter are not infinite, like the one I derived in my question but much simpler, as simple as original logistic map. $\endgroup$ – iLie May 14 at 13:58

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