Discrete Dynamical Systems

✍ Scribed by Oded Galor

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 159
• Category: Library

✦ Synopsis

This book provides an introduction to discrete dynamical systems – a framework of analysis that is commonly used in the ?elds of biology, demography, ecology, economics, engineering, ?nance, and physics. The book characterizes the fundamental factors that govern the quantitative and qualitative trajectories of a variety of deterministic, discrete dynamical systems, providing solution methods for systems that can be solved analytically and methods of qualitative analysis for those systems that do not permit or necessitate an explicit solution. The analysis focuses initially on the characterization of the factors that govern the evolution of state variables in the elementary context of one-dimensional, ?rst-order, linear, autonomous systems. The f- damental insights about the forces that a?ect the evolution of these - ementary systems are subsequently generalized, and the determinants of the trajectories of multi-dimensional, nonlinear, higher-order, non- 1 autonomous dynamical systems are established. Chapter 1 focuses on the analysis of the evolution of state variables in one-dimensional, ?rst-order, autonomous systems. It introduces a method of solution for these systems, and it characterizes the traj- tory of a state variable, in relation to a steady-state equilibrium of the system, examining the local and global (asymptotic) stability of this steady-state equilibrium. The ?rst part of the chapter characterizes the factors that determine the existence, uniqueness and stability of a steady-state equilibrium in the elementary context of one-dimensional, ?rst-order, linear autonomous systems.

✦ Table of Contents

Acknowledgements
Preface
Contents
1 One-Dimensional, First-Order Systems
1.1 Linear Systems
1.1.1 Characterization of the Solution
1.1.2 Existence of Steady-State Equilibria
1.1.3 Uniqueness of Steady-State Equilibria
1.1.4 Stability of Steady-State Equilibria
1.2 Nonlinear Systems
1.2.1 The Solution
1.2.2 Existence, Uniqueness and Multiplicity of Steady-State Equilibria
1.2.3 Linearization and Local Stability of Steady-State Equilibria
1.2.4 Global Stability
2 Multi-Dimensional, First-Order, Linear Systems: Solution
2.1 Characterization of the Solution
2.2 Existence and Uniqueness of Steady-State Equilibria
2.3 Examples of Two-Dimensional Systems
2.3.1 Explicit Solution and Stability Analysis
2.3.2 Stability Analysis Without an Explicit Solution
2.4 Properties of the Jordan Matrix
2.5 Representation of the System in the Jordan Normal Form
2.5.1 Transformation of Non-Homogeneous Systems into Homogeneous Ones
2.5.2 The Solution in Terms of the Jordan Normal Form
3 Multi-Dimensional, First-Order, Linear Systems: Characterization
3.1 Distinct Real Eigenvalues
3.1.1 Characterization of the Solution
3.1.2 Phase Diagrams of Two-Dimensional Uncoupled Systems
3.2 Repeated Real Eigenvalues
3.2.1 Characterization of the Solution
3.2.2 Phase Diagram of the Two-Dimensional Case
3.3 Distinct Pairs of Complex Eigenvalues
3.3.1 Characterization of the Solution
3.3.2 Phase Diagram of a Two-Dimensional System
3.4 Repeated Pairs of Complex Eigenvalues
3.5 The General Case
3.6 Characterization of Two-Dimensional Systems in Terms of tr A and det A
4 Multi-Dimensional, First-Order, Nonlinear Systems
4.1 Local Analysis
4.1.1 Linearization
4.1.2 Stable, Unstable, and Center Eigenspaces
4.1.3 Local Stable and Unstable Manifolds
4.1.4 The Stable Manifold Theorem
4.2 Global Analysis
5 Higher-Order and Non-Autonomous Systems
5.1 Higher-Order Systems
5.1.1 Linear Systems
5.1.2 Nonlinear Systems
5.2 Non-Autonomous Systems
6 Examples of Two-Dimensional Systems
6.1 First-Order Linear Systems
6.1.1 Real, Distinct, Positive Eigenvalues
6.1.2 Complex Eigenvalues Periodic Orbit
6.1.3 Complex Eigenvalues Spiral Sink
6.2 Second-Order Linear Systems
6.3 Nonlinear Systems
Glossary
References
Index