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Fernández-Guasti, M.

Imaginary scators bound set under the iterated quadratic mapping in \(1+2\) dimensional parameter space. (English) Zbl 1334.37044

Int. J. Bifurcation Chaos Appl. Sci. Eng. 26, No. 1, Article ID 1630002, 17 p. (2016).
Summary: The quadratic iteration is mapped within a nondistributive imaginary scator algebra in \(1+2\) dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional \(\mathbf{S}\) set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The \(\mathbf{S}\) set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period \(m\), are shown to be necessarily surrounded by points that produce a divergent magnitude after \(m\) iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the \(m\)th iteration, have preperiod 1 and period \(m\). Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary.

Cited in 3 Documents

MSC:

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)

Keywords:

3D bifurcations; hypercomplex numbers; imaginary scators; quadratic iteration; Mandelbrot set; discrete dynamical systems

Software:

GitHub; Gestaltlupe
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References:

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