Binary Coding Functions for Generalized Logistic Maps with z-Unimodality

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This Demonstration shows binary coding functions for one-dimensional iterative maps with -unimodality [1]. The test map, , generalizes the classic logistic map, [12–15]. Here is the iteration number, is the iterate of starting from the initial value (i.e. ), is the main control parameter, and is the subcontrol parameter ( determines the unimodality, the degree of the local maximum of ).


This Demonstration uses one of three coding functions




where is a unit-step function satisfying

and is another step function (a sign function) satisfying


The blue box on the left is a Locator that you can drag; the image inside the box is rescaled on the right in accordance with the zoom-in level . The binary coding function (, , or ) acts on 251 (or 2001) equally spaced initial condition for both plots.


Contributed by: Ki-Jung Moon (February 2014)
Based on programs by: Stephen Wolfram, Ed Pegg Jr, Enrique Zeleny, C. Pellicer-Lostao, and R. Lopez-Ruiz
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.

By dragging the sliders and the Locator, confirm and answer the following:

• the limit exists for any , , and and .

• For any and , the take on only values, so that is a full binary space.

and are fractal codes while is a monotonically increasing step function.

• Beyond the boundary crisis at (i.e. ), becomes a devil's staircase function. However, the devil's staircase shown in this Demonstration is different from that of the sine-circle map at the edge of chaos [16]. What are the differences between the two? What causes those differences?

• At , becomes a smooth function in for any . Why?

For or for , the iterates of rapidly approach (more rapidly for larger values of and ) and therefore, due to finite-precision arithmetic, 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 [1]. 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 .

Complex dynamical systems (CDS), such as the Mandelbrot set and Julia sets, are described by information theory and are therefore computable [1–3]. The existence of interesting two-color pixel division games for CDS 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) [3]. In symbolic dynamics, these are called invariant coordinates (or invariant descriptors) [4]. The same is true for dynamical systems in the real domain [4–7]. The related binary coding functions of controlled chaotic orbits can be used for encoding digital information [8–10]. The classic logistic map is a prototypical example of such systems, with many interesting key features in chaos communications [10–12]. See the Wikipedia articles for "Mandelbrot Set", "Julia Set", "Symbolic Dynamics", "Control of Chaos", "Logistic Map", and "Chaos Communications".


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

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

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

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

[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] S. H. Strogatz, Nonlinear Dynamics and Chaos, New York: Perseus Books Publishing, 1994.

[7] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, New York: Springer, 1996.

[8] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Boulder: Westview Press, 2003.

[9] E. Ott, C. Grebogi, and J. A. Yorke, "Controlling Chaos," Physical Review Letters, 64(11), 1990 pp. 1196–1199. doi:10.1103/PhysRevLett.64.1196.

[10] S. Hayes, C. Grebogi, and E. Ott, "Communicating with Chaos," Physical Review Letters, 70(20), 1993 pp. 3031–3034. doi:10.1103/PhysRevLett.70.3031.

[11] Y. C. Lai, "Encoding Digital Information Using Transient Chaos," International Journal of Bifurcation and Chaos, 10(4), 2000. doi:10.1142/S0218127400000554.

[12] M. J. Feigenbaum, "Quantitative Universality for a Class of Nonlinear Transformations," Journal of Statistical Physics, 19(1), 1978 pp. 25–52. doi:10.1007/BF01020332.

[13] M. J. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformations," Journal of Statistical Physics, 21(6), 1979 pp. 669–706. doi:10.1007/BF01107909.

[14] K.-J. Moon and S. D. Choi, "Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," Chaos: An Interdisciplinary Journal of Nonlinear Science, 18(2), 2008 pp. 023104. doi:10.1063/1.2902826.

[15] 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. doi:10.1063/1.3530128.

[16] M. H. Jensen, P. Bak, and T. Bohr, "Complete Devil's Staircase, Fractal Dimension, and Universality of Mode-Locking Structure in the Circle Map," Physical Review Letters, 50(21), 1983 pp. 1937-1939. doi:10.1103/PhysRevLett.50.1637.

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