Discrete Dynamical Systems (eBook)

Oded Galor (Autor)

eBook Download: PDF

2007
XIV, 153 Seiten
Springer Berlin (Verlag)
978-3-540-36776-5 (ISBN)

Infos

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.

Inhalt

Acknowledgements 5
Preface 6
Contents 9
1 One-Dimensional, First-Order Systems 11
1.1 Linear Systems 12
1.1.1 Characterization of the Solution 12
1.1.2 Existence of Steady-State Equilibria 14
1.1.3 Uniqueness of Steady-State Equilibria 15
1.1.4 Stability of Steady-State Equilibria 16
1.2 Nonlinear Systems 26
1.2.1 The Solution 26
1.2.2 Existence, Uniqueness and Multiplicityof Steady-State Equilibria 27
1.2.3 Linearization and Local Stabilityof Steady-State Equilibria 29
1.2.4 Global Stability 34
2 Multi-Dimensional, First-Order, Linear Systems: Solution 37
2.1 Characterization of the Solution 38
2.2 Existence and Uniqueness of Steady-State Equilibria 40
2.3 Examples of Two-Dimensional Systems 43
2.3.1 Explicit Solution and Stability Analysis 43
2.3.2 Stability Analysis Without an Explicit Solution 56
2.4 Properties of the Jordan Matrix 61
2.5 Representation of the System in the JordanNormal Form 66
2.5.1 Transformation of Non-Homogeneous Systemsinto Homogeneous Ones 66
2.5.2 The Solution in Terms of the JordanNormal Form 67
3 Multi-Dimensional, First-Order, Linear Systems: Characterization 69
3.1 Distinct Real Eigenvalues 69
3.1.1 Characterization of the Solution 69
3.1.2 Phase Diagrams of Two-Dimensional Uncoupled Systems 72
3.2 Repeated Real Eigenvalues 78
3.2.1 Characterization of the Solution 78
3.2.2 Phase Diagram of the Two-Dimensional Case 81
3.3 Distinct Pairs of Complex Eigenvalues 87
3.3.1 Characterization of the Solution 87
3.3.2 Phase Diagram of a Two-Dimensional System 92
3.4 Repeated Pairs of Complex Eigenvalues 94
3.5 The General Case 96
3.6 Characterization of Two-Dimensional Systems in Terms of trA and detA 97
4 Multi-Dimensional, First-Order, Nonlinear Systems 102
4.1 Local Analysis 105
4.1.1 Linearization 105
4.1.2 Stable, Unstable, and Center Eigenspaces 107
4.1.3 Local Stable and Unstable Manifolds 110
4.1.4 The Stable Manifold Theorem 111
4.2 Global Analysis 113
5 Higher-Order and Non-Autonomous Systems 115
5.1 Higher-Order Systems 115
5.1.1 Linear Systems 115
5.1.2 Nonlinear Systems 119
5.2 Non-Autonomous Systems 120
6 Examples of Two-Dimensional Systems 122
6.1 First-Order Linear Systems 122
6.1.1 Real, Distinct, Positive Eigenvalues 122
6.1.2 Complex Eigenvalues - Periodic Orbit 133
6.1.3 Complex Eigenvalues - Spiral Sink 140
6.2 Second-Order Linear Systems 144
6.3 Nonlinear Systems 151

Produkt-Details

• Erscheint lt. Verlag: 17.5.2007
• Zusatzinfo: XIV, 153 p. 53 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Allgemeines / Lexika
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Unternehmensführung / Management
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre
• Schlagworte: Complexity • discrete dynamical systems • Dynamical Systems • Dynamische Systeme • Higher-order Systems • Non-linear Dynamical Systems
• ISBN-10: 3-540-36776-4 / 3540367764
• ISBN-13: 978-3-540-36776-5 / 9783540367765