Numerical Investigation of a Nonlinear, Ordinary Differential Equation System

Abstract

In recent years, it has been found that many complicated, high order systems display behavior similar to behavior of the logistical equation, Fm(X)=mX(1-X), when the equation is viewed as a dynamical system. A dynamical system is some st of equations that describes the state at time t from knowledge of the previous history of the system, i.e. the states leading up to t. Using the logistic, the state at time t+1, denoted X*+1, is found by taking X*, the state at time t, and substituting into the equation: x*+1=Fm(X*)=mX*(1-X*). No attempt has been made to solve the equations exactly, except in special cases. An exact solution would be most difficult if not impossible to obtain. This investigation is limited to studying the behavior of the one-dimensional case (i.e. the logistic equation).