Chyi-Lung Lin This email address is being protected from spambots. You need JavaScript enabled to view it.1 and Mon-Ling Shei1

1Department of Physics, Soochow University, Taipei, Taiwan 111, R.O.C.


Received: May 12, 2005
Accepted: March 21, 2006
Publication Date: March 1, 2007

Download Citation: ||  


We prove that the well-known logistic map, f(x) = μx(1 x), is topologically conjugate to the map f(x) = (2 - μ) x(1 - x). The logistic map thus has the same dynamics at parameter values μ and 2 - μ, and hence has the μ → 2 - μ symmetry in dynamics. To examine this symmetry, we study the (μ, s)n relation of fn , which is obtained by eliminating x from the equations fn (x) = x and s = dfn (x)/dx. We then obtain an equation directly relating μ and s for period-n point of f. We derive the (μ, s)n relation for period n = 1, 2, 3, and 4, and we show that the (μ, s)n relations are invariant under the transformation of μ → 2 - μ.

Keywords: Logistic Map, Topologically Conjugate, Symmetry, Invariant, Periodic Bifurcation


  1. [1] Denny Gulick, Encounters with Chaos, Mc-GrawHill, Inc., p. 110 (1992).
  2. [2] Steven, H., Strogaze, Nonlinear Dynamics and Chaos, Addition-Wesley Publishing Company (1994).
  3. [3] Robert, L., Devaney, An Introduction to Chaotic Dynamical Systems, 2nd-Edition, Addison-Wesley Publishing Company (1989).
  4. [4] Robert, L., Devaney, A First Course in Chaotic Dynamical Systems Theory and Experiment, AddisonWesley Publishing Company, The Advanced Book Program (1992).
  5. [5] Mitchell Feigenbaum, “Quantitative Universality for a Class of Nonlinear Transformations,” J. of Stat. Physics, Vol. 19, p. 25 (1978).
  6. [6] Mitchell Feigenbaum, “The Universal Metric Properties of Nonlinear Transformations,” J. of Stat. Physics., Vol. 1, p. 69 (1979).
  7. [7] Pierre Collet and Jean-Pierre Eckmann, Iterated Maps on The Interval As Dynamical Systems, Birkhauser Boston, (1980).
  8. [8] Edward Ott, Chaos in Dynamical Systems, Cambridge University Press (1993).
  9. [9] Alligood, K. T., Sauer, T. D. and Yorke, J. A., Chaos, Spring-Verlag, New York, Inc. (1997).
  10. [10] Ibid. 2, p. 363.
  11. [11] Saha, P. and Strogatz, S. H., “The Birth of Period Three,” Mathematics Magazine, Vol. 68, pp. 4347 (1995).
  12. [12] Hillman, A. P. and Alexanderson, G. L., Abstract Algebra, PWS Publishing Company, Boston, MA. U.S.A. (2000)


42nd percentile
Powered by  Scopus

SCImago Journal & Country Rank

Enter your name and email below to receive latest published articles in Journal of Applied Science and Engineering.