Two-Color Pixel Division Game for Generalized Logistic Maps with z-Unimodality

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The pixel division game (PDG) is an excellent example to illustrate the appearance of the Fibonacci sequence, the golden ratio, and Farey trees in complex dynamical systems (CDS), such as the Mandelbrot set and Julia sets [1–4]. The key idea is that CDS are described by information theory and are therefore computable. The existence of interesting 2-color PDG for CDS [1] enables regions of the complex plane to be encoded using coding theory with the binary digits 0 and 1 (or -1 and 1, or more symbolically, L and R). In symbolic dynamics, these are called invariant coordinates [5]. The same is true for dynamical systems in the real domain. The related coding functions of controlled chaotic orbits can be used for encoding information [6–7]. The classic logistic map [8–9], , is a prototypical example of such systems, with many interesting key features in chaotic communications [6–7,10]. The test map used in this Demonstration, , generalizes , [11–12].


This Demonstration uses either of two pixel division rules



where is the main control parameter of , is the subcontrol parameter (which determines the unimodality, the degree of the local maximum of ), is an iteration number, is the iteration of starting from the initial value , and , , is a step function satisfying


The two rules and are step-like coding functions for the iteration of ; they return only -1 or +1. Now one can use the two built-in graphic functions ContourPlot[, ⋯] and DensityPlot[, ⋯] with or as the input function , which is suitable for two-color rendering purposes. For ContourPlot, an additional option Contours->{0} is needed.


Contributed by: Ki-Jung Moon (November 2013)
Open content licensed under CC BY-NC-SA



This Demonstration is meant to help users (especially students) understand computable aspects of one-dimensional iterative maps with -unimodality.

Things to try:

(a) When , confirm that there are only two large square pixels. Think of them as being labeled with 0 or 1 (0 for "the bright pixel" and 1 for "the dark pixel"). Put the bit 0 or 1 after below the decimal point, 0.0 for the bright pixel and 0.1 for the dark pixel.

(b) When , confirm that there are now four pixels, now labeled 0.00, 0.01, 0.10, or 0.11.

(c) Increase the number of iterations and observe the changes.

(d) What happens beyond the boundary (for )? (Use "drag" or the horizontal zoom to see the region.)

(e) At , confirm that the size of all pixels along the axis approaches zero. Why?

(f) What happens in the binary space? This is a finite binary space with small intervals of the same size , as decreases to 1, that is, .

(g) Between the two rules and , which one is better? Why do you think so?

For or for , the iterates of rapidly approach (more rapidly for larger values of and ) and therefore, due to the finite length in precision, the numbers are too large to compute. This is one of the main problems in computer simulations of iterated maps, as Stephen Wolfram mentioned in his 2002 book [13]. To remove this unwelcome problem, the author has done a simple, topological surgery to the test map without losing essential topological properties, so that this Demonstration can compute exact values for almost all initial conditions within the tolerance of practical precision for any and for any .


[1] H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, 2nd ed., New York: Springer, 2004.

[2] R. L. Devaney, "The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence," The American Mathematical Monthly, 106(4), 1999 pp. 289–302.

[3] The first pictures of the Mandelbrot set were drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups. On March 1, 1980, at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, Benoit Mandelbrot first saw a visualization of the set. For more detailed information, see the Wikipedia article for "Mandelbrot Set".

[4] B. Mandelbrot, The Fractal Geometry of Nature, San Francisco: W. H. Freeman, 1982.

[5] J. Milnor and W. Thurston, "On Iterated Maps of the Interval," in Dynamical Systems (1986–87), College Park, MD (A. Dold and B. Eckmann, eds.) Berlin: Springer, 1988 pp. 465–563.

[6] Y. C. Lai, "Encoding Digital Information Using Transient Chaos," International Journal of Bifurcation and Chaos, 10(4), 2000.

[7] S. Hayes, C. Grebogi, and E. Ott, "Communicating with Chaos," Physical Review Letters, 70(20), 1993 pp. 3031–3034.

[8] M. J. Feigenbaum, "Quantitative Universality for a Class of Non-Linear Transformations," Journal of Statistical Physics, 19, 1978 pp. 25–52.

[9] M. J. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformations," Journal of Statistical Physics, 21, 1979 pp. 669–706.

[10] Chaos communications is an application field of chaos theory that is aimed to provide security in the transmission of messages via telecommunication channels. For more detailed information, see the Wikipedia article for "Chaos Communications".

[11] K.-J. Moon, "Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," Chaos: An Interdisciplinary Journal of Nonlinear Science, 18, 2008 pp. 023104.

[12] K.-J. Moon, "Erratum: Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," Chaos: An Interdisciplinary Journal of Nonlinear Science, 20, 2010 pp. 049902.

[13] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, 2002.

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