The Logistic Function Part III: A Period-Doubling Cascade

Abstract

In Chapter 7, we characterized the behavior of the parametrized family of functions h _r (x) = rx(1 − x) for 0 < r ≤ 3. By this time, the reader should find it easy to verify that h has, at most, two periodic points for these parameter values, both of which are fixed. All other real numbers are in the stable set of one of these points or the stable set of infinity. In Chapter 8 we demonstrated that there is a Cantor set in [0,1] on which h is chaotic when $r > 2 + \sqrt 5$. All real numbers not in the Cantor set are in the stable set of infinity. We stated without proof that this property also holds for $4 < r \leqslant 2 + \sqrt 5$. Finally, we note in Corollary 9.5 that h is chaotic on [0,1] when r = 4. The analysis of the behavior of h for parameter values between 3 and 4 remains.