Ordinary Differential Equations

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Stephen Wiggins, University of Bristol

Pub Date: 2017

ISBN 13:

Publisher: Independent

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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


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