1. Introduction to Dynamic Models 1.1 Six Examples of Input/Output Dynamics
1.1.1 Smallpox in Montreal
1.1.2 Spread of Disease Equations
1.1.3 Filling a Container
1.1.4 Head Impact and Brain Acceleration
1.1.5 Compartment models and pharmacokinetics
1.1.6 Chinese handwriting
1.1.7 Where to go for More Dynamical Systems
1.2 What This Book Undertakes
1.3 Mathematical Requirements
1.4 Overview
2 DE notation and types
2.1 Introduction and Chapter Overview 2.2 Notation for Dynamical Systems
2.2.1 Dynamical System Variables
2.2.2 Dynamical System Parameters
2.2.3 Dynamical System Data Configurations
2.2.4 Mathematical Background
2.3 The Architecture of Dynamic Systems
2.4 Types of Differential Equations
2.4.1 Linear Differential Equations
2.4.2 Nonlinear Dynamical Systems
2.4.3 Partial Differential Equations
2.4.4 Algebraic and Other Equations
2.5 Data Configurations
2.5.1 Initial and Boundary Value Configurations
2.5.2 Distributed Data Configurations 2.5.3 Unobserved or Lightly Observed Variables
2.5.4 Observational Data and Measurement Models
2.6 Differential Equation Transformations
2.7 A Notation Glossary
3 Linear Differential Equations and Systems
3.1 Introduction and Chapter Overview
3.2 The First Order Stationary Linear Buffer
3.3 The Second Order Stationary Linear Equation
3.4 The mth Order Stationary Linear Buffer
3.5 Systems of Linear Stationary Equations
3.6 A Linear System Example: Feedback Control
3.7 Nonstationary Linear Equations and Systems 3.7.1 The First Order Nonstationary Linear Buffer
3.7.2 First Order Nonstationary Linear Systems
3.8 Linear Differential Equations Corresponding to Sets of Functions
3.9 Green's Functions for Forcing Function Inputs
4 Nonlinear Differential Equations
4.1 Introduction and Chapter Overview
4.2 The Soft Landing Modification
4.3 Existence and Uniqueness Results
4.4 Higher Order Equations
4.5 Input/Output Systems
4.6 Case Studies
4.6.1 Bounded Variation: The Catalytic Equation 4.6.2 Rate Forcing: The SIR Spread of Disease System
4.6.3 From Linear to Nonlinear: The FitzHugh-Nagumo Equations
4.6.4 Nonlinear Mutual Forcing: The Tank Reactor Equations
4.6.5 Modeling Nylon Production
5 Numerical Solutions
5.1 Introduction
5.2 Euler Methods
5.3 Runge-KuttaMethods
5.4 Collocation Methods
5.5 Numerical Problems
5.5.1 Stiffness
5.5.2 Discontinuous Inputs
5.5.3 Constraints and Transformations
6 Qualitative Behavior
6.1 Introduction
6.2 Fixed Points
6.2.1 Stability
6.3 Global Analysis and Limit Cycles
6.3.1 Use of Conservation Laws
6.3.2 Bounding Boxes
6.4 Bifurcations
6.4.1 Transcritical Bifurcations
6.4.2 Saddle Node Bifurcations
6.4.3 Pitchfork Bifurcations
6.4.4 Hopf Bifurcations
6.5 Some Other Features
6.5.1 Chaos
6.5.2 Fast-Slow Systems
6.6 Non-autonomous Systems
6.7 Commentary
7 Trajectory Matching
7.1 Introduction
7.2 Gauss-Newton Minimization
7.2.1 Sensitivity Equations
7.2.2 Automatic Differentiation
7.3 Inference
7.4 Measurements on Multiple Variables
7.4.1 Multivariate Gauss-Newton Method
7.4.2 VariableWeighting using Error Variance
7.4.3 Estimating s2
7.4.4 Example: FitzHugh-NagumoModels
7.4.5 Practical Proble
About the Author: Jim Ramsay, PhD, is Professor Emeritus of Psychology and an Associate Member in the Department of Mathematics and Statistics at McGill University. He received his PhD from Princeton University in 1966 in quantitative psychology. He has been President of the Psychometric Society and the Statistical Society of Canada. He received the Gold Medal in 1998 for his contributions to psychometrics and functional data analysis and Honorary Membership in 2012 from the Statistical Society of Canada.
Giles Hooker, PhD, is Associate Professor of Biological Statistics and Computational Biology at Cornell University. In addition to differential equation models, he has published extensively on functional data analysis and uncertainty quantification in machine learning. Much of his methodological work is inspired by collaborations in ecology and citizen science data.
