Algebraic orbits on period-3 window for the logistic map

• Main Author: Fidelis A.J.*
• Publication Date: 2015
• Other Authors: Martins L.C.*
• Format: Article
• Language: eng
• Source: Repositório Institucional da Udesc
• dARK ID: ark:/33523/001300000ksjr
• Download full: https://repositorio.udesc.br/handle/UDESC/8277

© 2014, Springer Science+Business Media Dordrecht.Algebraic stable and unstable orbits are presented for the famous period-3 window of the logistic map (Formula presented.). It is exhibited the general polynomial that gives rise to both stable and unstable period-3 orbits. These orbits are shown for three different fixed control parameter values of r: at tangent bifurcation (birth), at super-stability and at ending pitchfork bifurcation (death) of the period-3 window. All orbits are exposed in two different ways: a sum of complex numbers (Formula presented.), as proposed by Gordon (Math Mag 69:118–120, 1996), and via Euler’s formula (Formula presented.). The algebraic expressions of (Formula presented.) and θ are given for each r value for both stable and unstable orbits, as well as their numerical values and the Lyapunov exponent. It is shown that a and (Formula presented.) are statistical quantities of the orbits.