Nonlinear Dynamics

, Volume 79, Issue 2, pp 1015–1021 | Cite as

Algebraic orbits on period-3 window for the logistic map

  • Antônio João Fidélis
  • Luciano Camargo Martins
Original Paper


Algebraic stable and unstable orbits are presented for the famous period-3 window of the logistic map \(x_{n+1}=rx_n(1-x_n)\). 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 \(x_i=a+bc+\overline{bc}\), as proposed by Gordon (Math Mag 69:118–120, 1996), and via Euler’s formula \(x_i=a+2|b|\cos (\theta )\). The algebraic expressions of \(a, b, c, |b|\) and \(\theta \) 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 \(|b|\) are statistical quantities of the orbits.


Logistic map Algebraic results Periodic orbits  Bifurcation Period-3 window 



A. J. Fidélis was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and thanks IFC for providing part-time research. L. C. Martins thanks UDESC for providing part-time research and computational support.


  1. 1.
    Sharkovskii, A.N.: Coexistence of cycles of a continuous map of the line into itself. Int. J. Bifurc. Chaos 05(05), 1263–1273 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Li, T.Y., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82(10), 985–992 (1975)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ulam, S.M., von Neumann, J.: On combinations of stochastic and deterministic processes. Bull. Am. Math. Soc. 53, 1120 (1947)Google Scholar
  4. 4.
    Saha, P., Strogatz, S.H.: The birth of period three. Math. Mag. 68(1), 42–47 (1995)CrossRefMATHGoogle Scholar
  5. 5.
    Bechhoefer, J.: The birth of period 3, revisited. Math. Mag. 69(2), 115–8 (1996)MathSciNetMATHGoogle Scholar
  6. 6.
    Lee, M.H.: Analytical study of the super-stable-cycle in the logistic map. J. Math. Phys. 50, 122702 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gordon, W.B.: Period three trajectories of the logistic map. Math. Mag. 69, 118–120 (1996)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Antônio João Fidélis
    • 1
    • 2
  • Luciano Camargo Martins
    • 3
  1. 1.Instituto Federal Catarinense – IFC Campus LuzernaLuzernaBrazil
  2. 2.Universidade do Estado de Santa Catarina – UDESCJoinvilleBrazil
  3. 3.Departamento de FísicaUniversidade do Estado de Santa Catarina –UDESCJoinvilleBrazil

Personalised recommendations