On the study of chaotic behavior using the logistic function

Abstract

The study uses the idea of logistic function to illustrate the presence of chaos as the cause of the absence of periodic formation. The logistic function is used in demonstrating, proving, and explaining Definition 6 and Theorem 1 through examples, tables, and figures. A system in recurrent behavior is describable when it is stable. However, chaotic behavior is seen when the system moves beyond periodic making it difficult to predict or describe the nature of the system. The WolframAlpha computational knowledge engine was used in obtaining the tables and the figures for the study. The study shows that when the parameter of the logistic function is at exactly 4 there is an uncorrelated behavior of the system indicating a new regime called chaos. Finally, the study shows that after successive iterations of the system there is no recurrent formation which is due to the system showing un-periodic, unstable, and uncorrelated.