A Reproducible Expository Analysis of the Logistic Map: Stability, Period Doubling, Conjugacy and Symbolic Dynamics
Joshua Owolabi Adeleke
Department of Applied Mathematics
Illinois Institute of Technology
Chicago, IL 60616, USA
jadeleke@hawk.iit.edu
Abstract
The logistic map is the standard first example in introductions to discrete dynamical systems, but its treatment is scattered across sources, and the numerical practice behind its familiar pictures is rarely documented. A self-contained account is assembled here. Complete derivations are given for the fixed-point and period-two stability windows, including the frequently misstated marginal parameter, where the fixed point is non-hyperbolic yet still attracting. Period-doubling thresholds are determined to 10 significant figures by a simultaneous Newton solve, and ratios formed from four-decimal thresholds are shown incapable of converging to the Feigenbaum constant. A conjugacy with the tent map is proved and combined with the itinerary map to establish that the fully developed case is chaotic in the sense of Devaney, with topological entropy and Lyapunov exponent both equal to the natural logarithm of two. The numerical tables are independently cross-checked against exact formulas, stability criteria or higher-precision computations.
Keywords: logistic map; chaos; period doubling; Feigenbaum constant; Lyapunov exponent; tent map; symbolic dynamics; reproducible numerics
Cite this publication as:
J. O. Adeleke, “A Reproducible Expository Analysis of the Logistic Map: Stability, Period Doubling, Conjugacy and Symbolic Dynamics,” Complex Systems, 35(3), 2026 pp. 203–235.
https://doi.org/10.25088/ComplexSystems.35.3.203
