Study of a family of period three functions and its chaotic dynamics
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Abstract
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.
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