# Ordinary Differential Equations

Stephen Wiggins, University of Bristol

Copyright Year: 2017

Publisher: Stephen Wiggins

Language: English

## Formats Available

## Conditions of Use

Attribution

CC BY

## Reviews

Elegant and relatively short textbook is written on less than 150 pages but covers a 12-week course in 10 chapters and 6 appendices. The appendices take about a quarter of the book and could serve as review materials or lessons on their own. read more

Elegant and relatively short textbook is written on less than 150 pages but covers a 12-week course in 10 chapters and 6 appendices. The appendices take about a quarter of the book and could serve as review materials or lessons on their own.

In the Preface the author claims that he uses this textbook for the first course of ordinary differential equations for mathematics students, but it seems that this material is suitable for the second course. The book does not furnish proofs of theorems, but each chapter contains problem sets and few examples. The book contains the list of contents, biography, list of figures, list of tables, and index. Additional comments are provided on the margins.

For most US institutions the book appears to be an excellent complementary textbook for those who would like to gently introduce a little bit of mathematical theory but do not aim for too heavy load of theorems and definitions. Majority of the book is devoted to discussion about stability of two-dimensional autonomous systems. The emphasis is on the concept rather than calculations. The text is generously furnished with pictures. The textbook will remain valuable regardless the flow of time.

The textbook is extremely well written but with some overuse of language. Majority of the material is devoted to analysis of the stability of autonomous systems in two variables. The word “manifolds” used in the titles of chapters does not reflect on the generality used in the text that limits examples to curves on the plane. The manifold that actually appears in the textbook is a plane curve. The book contains the list of contents, biography, list of figures, list of tables, and index. It is easy to navigate through and the comments on the margins provide suggestions about the interconnections of topics.

All terms related to differential equations used in the textbook are introduced in a form of a definition. Many examples are assisted by pictures which significantly improve the clarity of the exposition. Notation, terminology and appearance are consistent throughout the book.

The book is conveniently divided into 10 chapters and 6 appendices with material carefully selected into a logical flow. Chapters do not carry additional subdivisions except steps of procedures, examples or sets of problems.

The content of the book is organized into a logical flow and contains a significant amount of cross references provided by comments on the generous margins. Each chapter begins with definitions, followed by examples or steps of procedures and ends with sample problems.

The interface is well organized according to TeX standards of books with broad margins left for comments. The author uses the margins skillfully providing cross references, footnotes, references and additional comments.

The text is grammatically excellent.

The book topic does not touch sensitive topics of race, culture, religion, background.

Suggestions for the textbook: few suggested reflexive problems for consideration may make a use of the vast margins. For example: in chapter 1 the author defines autonomous equations. In chapter 2 he proves that they have the time-shift property. A sharp student may develop curiosity whether every ODE having the time-shift property must necessarily be autonomous and how the proof may be approached in an elementary way using methods presented in the book.

## Table of Contents

- Preface
- 1 Getting Started: The Language of ODEs
- 2 Special Structure and Solutions of ODEs
- 3 Behavior Near Trajectories and Invariant Sets: Stability
- 4 Behavior Near Trajectories: Linearization
- 5 Behavior Near Equilbria: Linearization
- 6 Stable and Unstable Manifolds of Equilibria
- 7 Lyapunov's Method and the LaSalle Invariance Principle
- 8 Bifurcation of Equilibria, I
- 9 Bifurcation of Equilibria, II
- 10 Center Manifold Theory
- A Jacobians, Inverses of Matrices, and Eigenvalues
- B Integration of Some Basic Linear ODEs
- C Finding Lyapunov Functions
- D Center Manifolds Depending on Parameters
- E Dynamics of Hamilton's Equations
- F A Brief Introduction to the Characteristics of Chaos
- Bibliography
- Index

## About the Book

This book consists of ten weeks of material given as a course on ordinary differential equations (ODEs) for second year mathematics majors at the University of Bristol. It is the first course devoted solely to differential equations that these students will take.

This book consists of 10 chapters, and the course is 12 weeks long. Each chapter is covered in a week, and in the remaining two weeks I summarize the entire course, answer lots of questions, and prepare the students for the exam. I do not cover the material in the appendices in the lectures. Some of it is basic material that the students have already seen that I include for completeness and other topics are "tasters" for more advanced material that students will encounter in later courses or in their project work. Students are very curious about the notion of chaos, and I have included some material in an appendix on that concept. The focus in that appendix is only to connect it with ideas that have been developed in this course related to ODEs and to prepare them for more advanced courses in dynamical systems and ergodic theory that are available in their third and fourth years.

## About the Contributors

### Author

**Stephen Ray Wiggins** is an American applied mathematician, born in Oklahoma City, Oklahoma and best known for his contributions in nonlinear dynamics, chaos theory and nonlinear phenomena, influenced heavily by his PhD advisor Philip Holmes, whom he studied under at Cornell University. He is actively working on the advancement of computational applied mathematics at the University of Bristol, where he was the head of the Mathematics Department until 2008. Previously he was a professor at Caltech in Pasadena, California