The Nature of the Logistic Function as a Nonlinear Discrete Dynamical System
The logistic equation is a model of population growth rst introduced by Pierre- Francios Verhulst. This model is a continuous form that depends on time and a possible way of restructuring this continuous form of the equation into a discrete equation is known as the logistic map. The discrete logistic equation as a model is written as: xn+1 = rxn(1xn) where n = 0; 1; 2; 3:::; xn is the state at the discrete time n andr is the control parameter which operate within any given range and as a very simple example for nonlinear map in dynamics, it changes in behavior moving from one regime to another regime depends on the adjustment or variation of the control parameter. For some parameter values of r the logistic map display periodic behaviour (period-1 orbits \ xed point", period-2 orbits and period-n orbits), and for others, it displays chaotic behavior. This research seeks to explore and look into the behavior of the equation as r the parameter keeps increasing within a speci ed interval 1 to 4 inclusive [1, 4]. A geometrical procedure to examine the logistic function behavior through graphical analysis irrespective of the variations of the parameter r gives a pictorial nature of the map. In the displayed of the logistic map into periodic-orbits points through iterating, it shows the attracting and repelling behavior which depend on the parameter values. Beyond period-2 orbits are various kinds of period doubling that shows the logistic map behavior from the periodic regime into chaotic. Diagrammatically, bifurcation as commonly used in nonlinear dynamics gives a better behavior of the logistic map in dynamics as the control parameter r is varied. So, ideally in this research I seek to determine each periodic solution of range of the parameter r values by performing analysis for the periodic orbits so as to get a very good understanding of the bifurcation that are encountered in the logistic map. Chaotic regime as the nal regime of the logistic map as observed in the working lies on the increase of the parameter r within a certain range or for a particular value. Through higher periodic oscillating unstableness occurs leading to chaos, and it shows that the nal behavior of the logistic map is the chaotic regime. In studying the logistic map as a good and a perfect model into chaos, the concepts of iteration and orbits was also studied carefully as a foundation to the build-up of the main work. Work on iterating function and orbits were studied vigorously. Hence concluding that iterations occur when a particular function is evaluated over and over with its outputs starting with an initial inputs, the outputs obtained from each evaluation form the orbits in sequential order.
A thesis submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology in partial fulfillment of the requirement for the Degree of Master of Philosophy in Pure Mathematics, 2016