Using the Logistic Function to Illustrate Periodic Orbits as Recurrent Formation

Abstract

Recurrence as behaviour in the dynamical system is also a state which happens as a result of an outcome of a point which does return to its initial state. But in dynamical system the most simple and easy way to study regime in one dimension is the logistic function. In this study we seek to understudy the recurrence as a full strong state from the nature or behaviour of the logistic function, where the periodic orbits as behaviour the logistic function is considered. A point can only be termed as recurrent if it is in its own future state.  A periodic orbit returns infinitely often to each point on the orbit. And so it is clear that an orbit is recurrent when it returns repeatedly to each neighbourhood of its initial position. Recurrence as in dynamical system is a result of periodic formation which is a movement that returns back to the original state or position at a constant rate. A systematic example for each periodic point from the logistic function relative to a control parameter λ is discussed. Different iterations tables, diagrams (graphs) for xn  against n, tables of stability of periodic nature which shows relative range of the control parameter λ and x0  are discussed. Through graphical illustrations and algebraic approach, the study showed that in the formation of recurrence through logistic function, the parameter λ played a major role and not all the periodic points (orbits) lead to recurrent formation. The study also showed that unstable behaviour of the logistic function when λ=4, ends the periodic behaviour, hence the absence of recurrence. And the absence of recurrent and the unstable nature of system bring about chaos.