Topological Conjugacy of Logistic Map
Keywords:
Logistic map, Topological conjugacy, Topological transitivity, HomeomorphisimAbstract
Topological conjugacy’s preserve many topological dynamical properties. Thus, if we find a topological conjugacy of a map f with a simpler map g, we can analyse the simpler map g to obtain information about dynamical properties of the original map. In this paper we have established a topological conjugacy of the logistic map Lμ(x)=μx(1−x), x ∈ [0,1] with the quadratic map Q(x)=x2+c, tent map T: [0,1]→[0,1], T(x)={2x, if 0≤x≤1/2, and the logistic-type map Fμ(x)=(2−μ) x (1−x), x ∈ [0,1].
References
Denny Gulick, Encounters with Chaos, McGraw-Hill, Inc., p.110, 1992.
Steven H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley Publishing Company, 1994.
Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Addison-Wesley Publishing Company, 1989.
Robert L. Devaney, A First Course in Chaotic Dynamical Systems – Theory and Experiment, Addison-Wesley Publishing Company, The Advanced Book Program, 1992.
Mitchell Feigenbaum, “Quantitative Universality for a Class of Nonlinear Transformations,” Journal of Statistical Physics, Vol. 19, p. 25, 1987.
Mitchell Feigenbaum, “The Universal Metric Properties of Nonlinear Transformations,” Journal of Statistical Physics, Vol. 1, p. 69, 1979.
Pierre Collet and Jean-Pierre Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser Boston, 1980.
Edward Ott, Chaos in Dynamical Systems, Cambridge University Press, 1993.
Alligood, K. T., Sauer, T. D., and Yorke, J. A., Chaos, Springer-Verlag, New York, Inc., 1997.
Hilborn, R. C., Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, Oxford University Press, 1994.
Indranil Bhaumik and Binayak S. Choudhury, “Totally Transitive Maps – A Short Note,” General Notes, Vol. 3, No. 2, April 2011, pp. 80–83, ISSN 2219-7184, Copyright © ICSRS Publication, 2011.
M. Vellekoop and R. Berglund, “On Interval Transitivity = Chaos,” American Mathematical Monthly, 101(4), 1994, pp. 353–355.
I. Bhaumik and B. S. Choudhury, “Topologically Conjugate Maps and ωomegaω-Chaos in Symbol Space,” International Journal of Applied Mathematics, 23(2), 2010, pp. 309–321.
Geoffrey R. Goodson, Dynamical Systems, Chaos and Fractal Geometry.
Downloads
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
