Nonlinear Dynamics

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