One-Dimensional, First-Order Systems

Abstract

This chapter analyzes the evolution of a state variable in one-dimensional , first-order, discrete dynamical systems. It introduces a method of solution for these systems, and it characterizes the trajectory of the state variable, in relation to its steady-state equilibrium, examining the local and global (asymptotic) stability of this steady-state equilibrium. The first part of the chapter characterizes the factors determining the existence, uniqueness and stability of a steady-state equilibrium in the elementary context of one-dimensional, first-order, linear autonomous systems. Although linear dynamical systems do not necessarily govern the evolution of the majority of the observed dynamic phenomena, they serve as an important benchmark in the analysis of the qualitative properties of nonlinear systems, providing the characterization of the linear approximation of nonlinear systems in the proximity of steady-state equilibria. The second part of the chapter examines the trajectories of nonlinear systems based on the characterization of the linearized system in the proximity of a steady-state equilibrium. The basic propositions derived in this chapter provide the conceptual foundations for the generalization of the analysis and the characterization of multi-dimensional, higher-order, non-autonomous, dynamical systems. The qualitative analysis of these dynamical systems is based upon the examination of the factors that determine the actual trajectory of the state variable. However, as will become apparent, once the basic propositions that characterize the properties of these systems are derived , an explicit solution is no longer required in order to characterize the nature of these dynamical systems.