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Logistic map is a dynamic system based on real world processes in nature. Thus, it is possible to assign a meaning to multiplicands r, x, 1-x.

Using that, it is possible to construct abstract processes analogical to logmap and based on some mathematical structures, like maybe graph theory.

Are there some works on such topic or similar?

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To be honest, I don't have a clue about graph theory but generally speaking about the logistic map: As described by Metropolis et al. (On Finite Limit Sets for Transformations on the Unit Interval - 1973), every iterated map of the form $$x_{n+1}=r f(x_n) \ , \ with \ f(0)=f(1)=0$$ shows a behaviour similar to the logistic map. As $r$ is varied, the order in which stable periodic solutions appear is $independent$ of the unimodal map being iterated. Actually, this implies that the algebraic form of $f(x)$ is irrelevant, only its overall shape matters. The periodic attractors always occur in the same sequence, often referred to as the U-sequence. Some examples of this behaviour are the Belousov-Zhabotinsky chemical reaction (Simoyi et al. - 1982) or the Rossler Attractor. So as a general rule, any unimodal map of the form $x_{n+1}=r f(x_n)$ satisfying the conditions analyzed by Metropolis et al. can be studied on a similar way as the logistic map.

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