Periodic Point as a Recurrent Formation Using the Logistic Function: A Survey

Abstract

In topological dynamical systems (TDS), recurrence (periodic–like recurrence) is one of the important concepts in its studies but one major problem is the inability to demonstrate and/or illustrate its formation in the orbit structure of system from a topological point of view. In this paper, the logistic function was applied to demonstrate the periodic point as a recurrent formation (periodic–like recurrence) in the topological dynamical system and dynamical system. The Wolfram Alpha computational knowledge engine was used in obtaining the tables and the figures for the study through various examples of the logistic function. The study shows that period – 2 recurrence is formed when the trajectory of the function is made up of two different values that keep repeating after successive iterations as a result of the period – 2 orbits when the parameter of the function is between 3 and 3.45. The study again shows that when the parameter of the function is greater than 3.83 there is a period – 3 point hence the formation of other periodic points. Convincingly, beyond this period – 3 is another subsequent period called the period-doubling cascade leading into chaos. This period-doubling asserts that other periodic–like recurrences are also present, hence period –n  recurrent exists.