Experimental study of the dynamics of single and connected in a lattice complex-valued mappings: the architecture and interface of author’s software for modeling

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The paper describes a free software for research in the field of holomorphic dynamics based on the computational capabilities of the MATLAB environment. The software allows constructing not only single complex-valued mappings, but also their collectives as linearly connected, on a square or hexagonal lattice. In the first case, analogs of the Julia set (in the form of escaping points with color indication of the escape velocity), Fatou (with chaotic dynamics highlighting), and the Mandelbrot set generated by one of two free parameters are constructed. In the second case, only the dynamics of a cellular automaton with a complex-valued state of the cells and of all the coefficients in the local transition function is considered. The abstract nature of object-oriented programming makes it possible to combine both types of calculations within a single program that describes the iterated dynamics of one object.

The presented software provides a set of options for the field shape, initial conditions, neighborhood template, and boundary cells neighborhood features. The mapping display type can be specified by a regular expression for the MATLAB interpreter. This paper provides some UML diagrams, a short introduction to the user interface, and some examples.

The following cases are considered as example illustrations containing new scientific knowledge:

1) a linear fractional mapping in the form $Az^{n} +B/z^{n} $, for which the cases $n=2$, $4$, $n>1$, are known. In the portrait of the Fatou set, attention is drawn to the characteristic (for the classical quadratic mapping) figures of <>, showing short-period regimes, components of conventionally chaotic dynamics in the sea;

2) for the Mandelbrot set with a non-standard position of the parameter in the exponent $z(t+1)\Leftarrow z(t)^{\mu } $ sketch calculations reveal some jagged structures and point clouds resembling Cantor's dust, which are not Cantor's bouquets that are characteristic for exponential mapping. Further detailing of these objects with complex topology is required.

Keywords: holomorphic dynamics, nonlinear dynamics, cellular automata, experimental mathematics, fractal, MATLAB, iterated mappings
Citation in English: Matyushkin I.V., Rubis P.D., Zapletina M.A. Experimental study of the dynamics of single and connected in a lattice complex-valued mappings: the architecture and interface of author’s software for modeling // Computer Research and Modeling, 2021, vol. 13, no. 6, pp. 1101-1124
Citation in English: Matyushkin I.V., Rubis P.D., Zapletina M.A. Experimental study of the dynamics of single and connected in a lattice complex-valued mappings: the architecture and interface of author’s software for modeling // Computer Research and Modeling, 2021, vol. 13, no. 6, pp. 1101-1124
DOI: 10.20537/2076-7633-2021-13-6-1101-1124

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