Discrete Dynamical Systems
Course: History of Mathematics
Mount Mercy University
Spring 2013 abstract This document contains introductory concepts that are necessary for the comprehension of the Theory of Chaos, which is the irregular and unpredictable time evolution of many nonlinear systems [2]. The main points of this document will discuss the discrete logistic equation, fixed points, and periodic points. This document will give background information about the subjects with definitions and detailed explanations of the formulas and figures. This document is the first of two, which will lead to more details about bifurcations, chaos, and chaos in the Lorenz system.
1. Introduction
This document is a continuation of my study of differential equations. Mainly, discrete dynamical systems, the topics listed below will be further explored in the following section. * The Discrete Logistic Equation * Exponential growth model * The logistic difference equation * Iteration and cycles * Fixed and Periodic Points * Attracting and Repelling fixed points * Classification of fixed points * Classification of periodic points
In this document we begin the study of a model for progressions that evolve over time, specifically discrete dynamical systems, or difference equations. As opposed to differential equations, these models are well fitted to circumstances in which changes occur at particular times, rather than continuously.
The study of discrete dynamical systems most typically involves the process of iteration. To iterate, means to repeat a procedure numerous times [2]. In discrete dynamics the procedure that is repeated is the application of a mathematical function. Those familiar with differential equations may have encountered the iterative process, as it was used in Euler’s method for solving a differential equation.
We will see that finding exact solutions of discrete systems is not often possible. We will rely on a combination of numerical, analytical, and graphical techniques to comprehend these systems. Although, we have these advanced techniques we are unable to entirely describe even one-dimensional, nonlinear iterations. The reason is that many discrete nonlinear systems behave in a complex and unpredictable manner. This phenomenon, called chaos by mathematicians, will be discussed in further work [2].
2. parts of the manuscript
2.1. Exponential Growth Model
The exponential growth model is a model that suggests that a population will grow exponentially through time until a ceiling is reached. Once this ceiling is reached, the population’s needs would exceed the resources. In view of this fact, the population’s growth turns into a problem. Since, it would be unlikely that the population’s growth would become zero, this model becomes unrealistic. Although, this is an unrealistic model it is simple and illustrates the main ideas behind discrete dynamics [4]. We will use this model to tell us what the next generation will be or any other later generation. We will do so by using the equation
Pn+1 = kPn (1) where Pn is denoted by the population of the species at the end of the nth time period and k is the constant that determines the growth rate. Pn+1 will tell us the following population’s growths, which will be poportional to Pn.
2.2. Logistic Difference Equation
The logistic difference equation is applied to show a more realistic change in the growth of a population by adding assumptions that account for overgrowth. The assumptions that are made are, * The population at the end of the next generation is proportional to the population at the end of the current generation when the population is very small. * If the population is too large, then all resources will be used and the entire population will die out in the next generation and extinction will result [2].
Thus the logistic difference equation is produced, Pn+1 = kPn(1- Pn)