Acta Mathematica Sinica, English Series

, Volume 34, Issue 5, pp 891–900 | Cite as

On the Periodic Logistic Map

  • Cui Ping Li
  • Ming Zhao


In this paper, the famous logistic map is studied in a new point of view. We study the boundedness and the periodicity of non-autonomous logistic map
$${x_{n + 1}} = {r_n}{x_n}\left( {1 - {x_n}} \right),n = 0,1, \ldots ,$$
where {r n } is a positive p-periodic sequence. The sufficient conditions are given to support the existence of asymptotically stable and unstable p-periodic orbits. This appears to be the first study of the map with variable parameter r.


Non-autonomous logistic map periodic orbit asymptotic stable 

MR(2010) Subject Classification

37B55 37C05 37E05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Boyce, M. S., Daley, D. J.: Population tracking of fluctuating environments and natural selection for tracking ability. Amer. Nature, 115, 480–491 (1980)CrossRefMathSciNetGoogle Scholar
  2. [2]
    Coleman, B. D., Hsich, Y. H., Knowles, G. P.: On the optimal choice of r for a population in a periodic environment. Math. Biosci, 46, 71–85 (1979)CrossRefMathSciNetGoogle Scholar
  3. [3]
    Gaut, G. R. J., Goldring, K., Grogan, F., et al.: Difference equations with the Allee effect and the periodic sigmoid Beverton-Holt equation revisited. J. Biological Dynamics, 6, 1019–1033 (2012)CrossRefGoogle Scholar
  4. [4]
    Jia, Y.: Robust control with decoupling performance for steering and traction of 4WS vehicles under velocity-varying motion. IEEE Trans. Control Systems Technology, 8(3), 554–569 (2000)CrossRefGoogle Scholar
  5. [5]
    Jia, Y.: Alternative proofs for improved LMI representations for the analysis and the design of continuoustime systems with polytopic type uncertainty: A predictive approach. IEEE Trans. Autom. Control, 48(8), 1413–1416 (2003)CrossRefzbMATHGoogle Scholar
  6. [6]
    Li, C. P.: Semigroup structures and properties of some kinds of mappings. Sci. Sin. Math., 47(1), 135–146 (2017)CrossRefGoogle Scholar
  7. [7]
    Robinson, C.: Dynamical Systems: satbility, symbolic dynamics, and chaos. 2nd edition, CRC Press LLC, Florida, 2000Google Scholar
  8. [8]
    Saber N. Elaydi, Robert J. Sacker: Global stability of periodic orbits of non-autonumous difference equations and population biology. J. Differential Equations, 208, 258–273 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    Saber N. Elaydi, Robert J. Sacker: Population models with Allee maps: a new model. J. Biological Dynamics, 4, 397–408 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Yang, Y., Robert J. Sacker: Periodic unimodal Allee maps, the semigroup peoperty and the λ-Ricker map with Allee effect. Discrete Continuous Dynam. Systems-B, 19, 589–606 (2014)CrossRefzbMATHGoogle Scholar
  11. [11]
    Zeidler, E.: Applied Functional Analysis. Appl. Math. Phys., Springer, New York, 1991Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.LMIB-School of Mathematics and Systems ScienceBeihang UniversityBeijingP. R. China

Personalised recommendations