"Logistic map": an analytical solution (1995)

An analytical solution for the well-known quadratic recursion, the logistic map,is presented. Our derivation is based on the analogy between this recursion and a probabilistic problem that can besolved analytically. Thesolution is represented as a power of a transfer matrix

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Explicit solutions for arbitrary $r$ exist in various forms:

• Logistic map: an analytical solution (1995) represents the solution as a power of a transfer matrix.

• An explicit solution for the logistic map (1999) gives a functional integral solution.

• A note on exact solutions of the logistic map (2020, paywall) gives the solution in terms of a power series.