文档介绍：Discrete Dynamical Systems
Oded Galor∗
Brown University
November 17, 2005
Abstract
This book analyzes the fundamental factors that govern the quantitative and qualita-
tive behavior of discrete dynamical systems. It introduces solution methods for discrete
dynamical systems that can be examined analytically, as well as methods of analysis for
discrete dynamical systems that can be solved analytically. The analysis focuses initially on
the derivation of basic propositions about the factors that governs the evolution of a state
variable in the elementary context of a one-dimensional, first-order, linear autonomous,
systems. The analysis of the linear system and its stability, is followed by the examination
of the non-linear system. These basic propositions are subsequently generalized to account
for solutions and stability analysis in multi-dimensional, higher-order, non-autonomous,
nonlinear, dynamical systems.
Keywords: Discrete Dynamical Systems, Difference Equations, Global Stability, Local
Stability, Non-Linear Dynamics, Stable Manifolds.
JEL Classification Numbers: C62, O40
∗The author wishes to thank Ruben Durante, Petros Milionis and Takuma Kunieda for ments.
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Table of Content
1. Introduction
2. One-Dimensional, Autonomous, First-Order Systems
. Linear Systems
. The Solution
. Existence of Stationary Equilibria
. Uniqueness of Stationary Equilibria
. Stability of Stationary Equilibria
. Nonlinear Systems
. The Solution
. Existence, Uniqueness and Multiplicity of Stationary Equilibria
. Linearization and Local Stability of Stationary Equilibria
. Global Stability
3. Multi-Dimensional, Autonomous, First-Order, Linear Systems
.. The Solution
. Existence and Uniqueness of Stationary Equilibria
. Examples of a 2-D System
. Explicit Solution and Stability Analysis
. The Construction of Phase Diagrams without an Explicit Solution