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Is there a way to solve a logistic map $$x_{n+1} = kx_n(1-x_n)$$ without using numerical methods/computers?


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As to explicit formulas, they becomes complicated with the $n$ and no wonder considering the properties of this map. In this question are some partial results for the coefficients of iterated series $f(f\ldots f(x)\ldots))\;$ for $f(x)=ax+bx^2+cx^3+\ldots$. Here $a=k$, $b=-k$ and all the others coefficients are equal to zero. For $k=2$ and $k=4$ the solutions can be written out explicitly.

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For special values of $k$ (in particular, when $k=-2$, $k=2$, and $k=4$), there are explicit expressions for the iterates in terms of trigonometric/exponential functions. On the other hand, you'll need a numerical method anyway to evaluate the logarithm/exponential/arccosine, so I don't really know what you're expecting out of a "non-numerical" method...

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