Nonlinear Dynamics

A hands-on introductory survey

By (author) Marc R Roussel

Publication date:

30 April 2019

Publisher

Morgan & Claypool Publishers

ISBN-13: 9781643274645

BOOK DESCRIPTION
TABLE OF CONTENTS

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.

Key features

Includes embedded reader exercises

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

Nonlinear Dynamics

A hands-on introductory survey

By (author) Marc R Roussel

Publication date:

Publisher

ISBN-13: 9781643274645

Key features

Ingram Publisher Services UK

Customer Services