Nonlinear Dynamics: A Hands-On Introductory Survey

Description

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. 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