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Characterization of a New Potential Family of Organic-Like Pattern-Generating Dynamical Systems

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Abstract

In this paper, we characterize a new broad family of discrete-time dynamical systems, whose organic-like patterns resemble part of the external morphology of some families of invertebrates and the bioluminescence of some specific families of zooplankton. We also present a new type of fractal structure that is hidden inside the relationship between the main control parameters of this kind of systems. The approach to obtain these patterns lies in observing that they are visualized exclusively when the families of systems are interpreted as a whole, unique, global structure, emerging an organic-like pattern only when they are assembled. The characterization is based on three pillars: facts obtained from numerical computation, the analysis of the bifurcation diagrams, and the study of the new fractal structure. The aforementioned new broad family of discrete-time dynamical systems and a new type of fractal structure are formulated. The study of these new systems could help in finding unexpected alternatives to the representation of the patterns generated by the mechanisms of segmentation and bioluminescence, especially in invertebrates.

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Notes

  1. The terms “discrete-time dynamical system” and “map” will be used indistinctly along this manuscript.

  2. In the recent literature, the works of Aubin and Dahan [8], and Holmes [9], have been very useful for us to understand the state of the art of the field.

  3. E.g., its application to game theory to simulate the biological strategy-oriented evolution of species, presented by Cases and Anchorena in Grana, Duro, d’Anjou and Wang’s book “Information Processing with Evolutionary Algorithms” [10].

  4. There are also interesting similarities in other fields of study, e.g., Christoforou et al. [12] have adapted Evolutionary Game Theory to master-working computing, defining a reliability algorithm for Internet-based master–worker task computations, whose iterative formula resembles some of the aforementioned population models described by May and Oster.

  5. Tested programatically by applying randomized initial conditions and visualizing the attractors.

  6. The patterns are shown in white color over a black background for a better observation of the resemblance to the external morphology of some families of invertebrates and the mechanisms of bioluminescence of some species. The remaining figure (Fig. 4, study of the bifurcation diagrams) is shown using the same rule for a better visualization of the local bifurcations.

  7. As stated in the Definition 2 of the present chapter, we have computationally verified that \({\text {Max}}D=8 \times 10^3\) is a good enough maximum value to observe the behavior of the discrete-time family of dynamical systems.

  8. This kind of escape time algorithm is used, for instance, to visualize the classical Gamma function fractal.

  9. There are other interesting studies about the fractal nature of the morphology of invertebrates. For instance Castrejón et al. [15] evaluated the geometrical complexity of butterflies’ wings through the calculation of the fractal dimension of their patterns.

  10. The patterns presented in this paper are just some initial examples. If an automation method is defined, e.g., using NCC (Normalized Correlation Coefficient) based techniques to match the resulting patterns with possible families of insects, new similarities should be found with a better time–effort cost.

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Acknowledgements

The author would like to thank the editor and the reviewers for their valuable suggestions and comments, which greatly improve the quality of the paper. Thanks are also due to Noemí Martín Santo and Celio H. Barreto for the very useful insights and corrections.

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    David M. Marciel

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Marciel, D.M. Characterization of a New Potential Family of Organic-Like Pattern-Generating Dynamical Systems. SN COMPUT. SCI. 1, 26 (2020). https://doi.org/10.1007/s42979-019-0028-6

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