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- W1810040905 abstract "We return now to the analysis of the logistic function. Our goal is to prove that if (r > 2 + sqrt 5 ), then hr(x) = rx(1 − x) is chaotic on Λ. We recall that Λ is the set of all numbers in [0, 1] which remain in [0, 1] under iteration of h. That is, Λ = {x | hn(x) is in [0, 1] for all n}. By Theorem 9.20, it is sufficient to show that the periodic points of h are dense in Λ and that h is topologically transitive on Λ. We have already shown that the periodic points of h are dense on Λ in exercise 8.4. Unfortunately, proving that h is topologically transitive on Λ directly from the definition is a relatively difficult task. Consequently, we will show instead that the dynamics of h on Λ are the same as the dynamics of σ on Σ2. Mathematically speaking, we say that h on Λ is topologically conjugate to σ on Σ2." @default.
- W1810040905 created "2016-06-24" @default.
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- W1810040905 date "1994-01-01" @default.
- W1810040905 modified "2023-09-27" @default.
- W1810040905 title "The Logistic Function, Part II: Topological Conjugacy" @default.
- W1810040905 doi "https://doi.org/10.1007/978-1-4684-0222-3_10" @default.
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