Discrete Dynamical Systems

This book provides an introduction to discrete dynamical systems - a framework of analysis that is commonly used in the fields of biology, demography, ecology, economics, engineering, finance, 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, first-order, linear, autonomous systems. The fundamental insights about the forces that affect the evolution of these elementary systems are subsequently generalized, and the determinants of the trajectories of multi-dimensional, nonlinear, higher-order, non-autonomous dynamical systems are established. Chapter 1 focuses on the analysis of the evolution of state variables in one-dimensional, first-order, autonomous systems. It introduces a method of solution for these systems, and it characterizes the trajectory 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 first 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, first-order, linear autonomous systems.

List of contents

One-Dimensional, First-Order Systems.- Multi-Dimensional, First-Order, Linear Systems: Solution.- Multi-Dimensional, First-Order, Linear Systems: Characterization.- Multi-Dimensional, First-Order, Nonlinear Systems.- Higher-Order and Non-Autonomous Systems.- Examples of Two-Dimensional Systems.

Summary

This book provides an introduction to discrete dynamical systems – a framework of analysis that is commonly used in the fields of biology, demography, ecology, economics, engineering, finance, 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, first-order, linear, autonomous systems. The fundamental insights about the forces that affect the evolution of these elementary systems are subsequently generalized, and the determinants of the trajectories of multi-dimensional, nonlinear, higher-order, non-autonomous dynamical systems are established. Chapter 1 focuses on the analysis of the evolution of state variables in one-dimensional, first-order, autonomous systems. It introduces a method of solution for these systems, and it characterizes the trajectory 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 first 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, first-order, linear autonomous systems.

Additional text

From the reviews:"Discrete dynamical systems are an interesting subject both for mathematicians and for applied scientists. This book is an introduction to this topic. It consists of 6 chapters. The first one focuses on the analysis of the evolution of state variables in one dimensional first-order autonomous linear systems. The second chapter develops the solutions for multidimensional first-order autonomous linear systems. … The presentation is clear and systematic. … it is for advanced undergraduate and graduate students in mathematics and applied scientists. e.g. economy, biology etc." (Ahmed Hegazi, Zentralblatt MATH, Vol. 1107 (9), 2007)"The book under review gives an elementary introduction to the theory of discrete dynamical systems. … The presentation is clear and systematic. … ‘the book is designed for advanced undergraduate and graduate students in the fields of demography, ecology, economics, engineering, evolutionary biology, finance, mathematics, and physics’ with some basic knowledge in linear algebra and differential calculus." (Christian Pötzsche, Mathematical Reviews, Issue 2008 c)“The book aims at giving an elementary introduction into some basics of discrete dynamical systems. … Very explicit explanations are provided … . The text is addressed mainly to students from economics.” (Hans Crauel, Zentralblatt MATH, Vol. 1223, 2011)

Report

From the reviews:

"Discrete dynamical systems are an interesting subject both for mathematicians and for applied scientists. This book is an introduction to this topic. It consists of 6 chapters. The first one focuses on the analysis of the evolution of state variables in one dimensional first-order autonomous linear systems. The second chapter develops the solutions for multidimensional first-order autonomous linear systems. ... The presentation is clear and systematic. ... it is for advanced undergraduate and graduate students in mathematics and applied scientists. e.g. economy, biology etc." (Ahmed Hegazi, Zentralblatt MATH, Vol. 1107 (9), 2007)
"The book under review gives an elementary introduction to the theory of discrete dynamical systems. ... The presentation is clear and systematic. ... 'the book is designed for advanced undergraduate and graduate students in the fields of demography, ecology, economics, engineering, evolutionary biology, finance, mathematics, and physics' with some basic knowledge in linear algebra and differential calculus." (Christian Pötzsche, Mathematical Reviews, Issue 2008 c)
"The book aims at giving an elementary introduction into some basics of discrete dynamical systems. ... Very explicit explanations are provided ... . The text is addressed mainly to students from economics." (Hans Crauel, Zentralblatt MATH, Vol. 1223, 2011)

Product details

• Authors: Oded Galor
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 16.12.2006
• EAN: 9783540367758
• ISBN: 978-3-540-36775-8
• No. of pages: 153
• Weight: 430 g
• Illustrations: XIV, 153 p. 53 illus.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Social sciences, law, business > Business > Economics; B, Dynamics, complexity, Economics and Finance, Economic Theory, Management science, Dynamical Systems and Ergodic Theory, Ergodic theory, Nonlinear science, Dynamical systems, Quantitative Economics, Economic Theory/Quantitative Economics/Mathematical Methods, Cybernetics & systems theory, Applied Dynamical Systems, Computational complexity