Two-Color Pixel Division Game for Generalized Logistic Maps with z-Unimodality
Two-Color Pixel Division Game for Generalized Logistic Maps with z-Unimodality
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], (λ,)==λ(+1), 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, (λ,)=≡λ(1-|)2-1, generalizes (λ,), [11–12].
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This Demonstration uses either of two pixel division rules
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where is the main control parameter of (λ,), is the subcontrol parameter (which determines the unimodality, the degree of the local maximum of (λ,)), is an iteration number, is the iteration of (λ,) starting from the initial value , and (λ,), , is a step function satisfying
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-1 | if | -∞< x j |
+1 | if | 0≤ x j |
The two rules and are step-like coding functions for the iteration of (λ,); 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.
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