This book uses 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. This book also provides a survey that can be taught in a single academic term covering a greater variety of dynamical systems (discrete versus continuous time, finite versus infinite-dimensional, dissipative versus conservative) than is 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.
Author(s): Marc R. Roussel
Series: IOP Concise Physics
Publisher: IOP Publishing
Year: 2019
Language: English
Pages: 190
City: Bristol
PRELIMS.pdf
Preface
Author biography
Marc R Roussel
CH001.pdf
Chapter 1 Introduction
1.1 What is a dynamical system?
1.2 The law of mass action
1.3 Software
Reference
CH002.pdf
Chapter 2 Phase-plane analysis
2.1 Introduction
2.2 The Lindemann mechanism
2.3 Dimensionless equations
2.4 The vector field
2.5 Exercises
CH003.pdf
Chapter 3 Stability analysis for ODEs
3.1 Linear stability analysis
3.2 Lyapunov functions
3.3 Exercises
Reference
CH004.pdf
Chapter 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
References
CH005.pdf
Chapter 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
Reference
CH006.pdf
Chapter 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
References
CH007.pdf
Chapter 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
References
CH008.pdf
Chapter 8 Hamiltonian systems
8.1 Introduction
8.2 Integrable systems
8.3 Numerical integration
8.4 Exercises
CH009.pdf
Chapter 9 Nonautonomous systems
9.1 Introduction
9.2 A driven Brusselator
9.3 Automated bifurcation analysis
9.4 Exercises
Reference
CH010.pdf
Chapter 10 Maps and differential equations
10.1 Numerical methods as maps
10.2 Solution maps of differential equations
10.3 Poincaré maps of systems with periodic nonautonomous terms
10.4 Poincaré sections and maps in autonomous systems
10.5 Next-amplitude maps
10.6 Concluding comments
10.7 Exercises
References
CH011.pdf
Chapter 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
References
CH012.pdf
Chapter 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
Reference
CH013.pdf
Chapter 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
APP1.pdf
Chapter
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
