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Entreciencias: diálogos en la sociedad del conocimiento

versão On-line ISSN 2007-8064

Resumo

MACIAS PONCE, Julio César  e  MARTINEZ ALVAREZ, Luis Fernando. Study of a family of period three functions and its chaotic dynamics. Entreciencias: diálogos soc. conoc. [online]. 2019, vol.7, n.19, pp.11-25.  Epub 11-Jun-2020. ISSN 2007-8064.  https://doi.org/10.22201/enesl.20078064e.2018.19.65822.

Purpose:

To build one-dimensional chaotic dynamical systems through the study of functions with domain and codomain in the interval [0, 1] which is defined in terms of four parameters.

Methodology:

Based on the parameters that define each function that is proposed, those which have period three were identified and which induce a chaotic system in the context of Li-Yorke. The fixed point and Sharkovskii theorems were the fundamental tools in this work.

Results:

We obtained a set of chaotic dynamic systems. In turn, we described a simple process in order to obtain chaotic dynamic systems (additional to those obtained) and we suggest, as a first application, the obtainment of pseudo-random numbers.

Limitations:

The dynamic systems that were built are chaotic in the Li-Yorke sense -not necessarily in the Devaney sense-.

Findings:

The functions that were studied have a Zeta form graphic, and for each of those we identified its respective dual (the obtained graphics present a symmetric relation) and that is how we show the conditions that must verify the parameters -primal and dual- in order to obtain (or not) period three.

Palavras-chave : Chaos; Sharkovskii; dynamic systems; orbit.

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