# Nonlinear Dynamics

- A hands-on introductory survey

- Marc R Roussel

- April 2019

### Description

This book employs a hands-on approach to nonlinear dynamics using commonly available software, including the free dynamical systems software Xppaut, Matlab (or its free cousin, Octave) and the Maple symbolic algebra system. Detailed instructions for various common procedures, including bifurcation analysis using the version of AUTO embedded in Xppaut, are provided. A survey that can be taught in a single academic term is provided, and it covers a greater variety of dynamical systems (discrete vs continuous time, finite vs infinite-dimensional, dissipative vs conservative) than are normally seen in introductory texts. Numerical computation and linear stability analysis are used as unifying themes throughout the book. Despite the emphasis on computer calculations, theory is not neglected, and fundamental concepts from the field of nonlinear dynamics, such as solution maps and invariant manifolds, are presented.### About Editors

Marc R Roussel earned a bachelor's degree in chemical physics from Queen's University in 1988. He then went on to graduate work in the Chemical Physics Theory Group at the University of Toronto under the supervision of Simon J Fraser, earning a MSc in 1990 and a PhD in 1994. In 1995, Roussel was hired as an assistant professor by the Department of Chemistry at the University of Lethbridge. He was promoted to associate professor in 2000, and to professor in 2005.### Table of Contents

1 Introduction

1.1 What is a dynamical system?

1.2 The Law of Mass Action

1.3 Software

Bibliography

2 Phase-Plane Analysis

2.1 Introduction

2.2 The Lindemann mechanism

2.3 Dimensionless equations

2.4 The vector field

2.5 Exercises

3 Stability Analysis for ODEs

3.1 Linear stability analysis

3.2 Lyapunov functions

3.3 Exercises

Bibliography

4 Introduction to bifurcations

4.1 Introduction

4.2 Saddle-node bifurcation

4.3 Transcritical bifurcation

4.4 Andronov-Hopf bifurcations

4.5 Dynamics in three dimensions

4.6 Exercises

Bibliography

5 Bifurcation Analysis with AUTO

5.1 Bifurcation diagram of a gene expression model

5.2 The phase diagram of Griffith's model

5.3 Bifurcation diagram of the autocatalator

5.4 Getting out of trouble in AUTO

5.5 Exercises

Bibliography

6 Invariant manifolds

6.1 Introduction

6.2 Flow dynamics away from the equilibrium point

6.3 Special eigenspaces of equilibrium points

6.4 From eigenspaces to invariant manifolds

6.5 Applications of invariant manifolds

6.5.1 The Lindemann mechanism revisited

6.5.2 A simple HIV model

6.6 Exercises

Bibliography

7 Singular perturbation theory

7.1 Introduction

7.2 Scaling and balancing

7.3 The outer solution

7.4 The inner solution

7.5 Matching the inner and outer solutions

7.6 Geometric singular perturbation theory and the outer solution

7.7 Exercises

Bibliography

8 Hamiltonian systems

8.1 Introduction

8.2 Integrable systems

8.3 Numerical integration

8.4 Exercises

9 Nonautonomous systems

9.1 Introduction

9.2 A driven Brusselator

9.3 Automated bifurcation analysis

9.4 Exercises

Bibliography

10 Maps and differential equations

10.1 Numerical methods as maps

10.2 Solution maps of differential equations

10.3 Poincar´e maps for nonautonomous systems

10.4 Poincar´e sections and maps in autonomous systems

10.5 Next-amplitude maps

10.6 Concluding comments

10.7 Exercises

Bibliography

11 Maps: Stability and bifurcation analysis

11.1 Linear stability analysis of fixed points

11.2 Stability of periodic orbits

11.3 Lyapunov exponents

11.4 Exercises

Bibliography

12 Delay-differential equations

12.1 Introduction to infinite-dimensional dynamical systems

12.2 Introduction to delay-differential equations

12.3 Linearized stability analysis

12.4 Exercises

Bibliography

13 Reaction-diffusion equations

13.1 Introduction

13.1.1 The rate of diffusion

13.1.2 Reaction-diffusion equations

13.2 Stability analysis of reaction-diffusion equations

13.3 Exercises

A Software Installation

A.1 Linux

A.1.1 XPPAUT

A.1.2 OCTAVE

A.2 Mac OS X

A.2.1 XPPAUT

A.2.2 OCTAVE

A.3 Windows

A.3.1 XPPAUT

A.3.2 OCTAVE

Index

### Bibliographic

Paperback ISBN: 9780750329835

Ebook ISBN: 9781643274638

DOI: 10.1088/2053-2571/ab0281

Publisher: Morgan & Claypool Publishers