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.
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© 1994 Springer-Verlag New York, Inc.
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Holmgren, R.A. (1994). The Logistic Function, Part II: Topological Conjugacy. In: A First Course in Discrete Dynamical Systems. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0222-3_10
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DOI: https://doi.org/10.1007/978-1-4684-0222-3_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94208-7
Online ISBN: 978-1-4684-0222-3
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