This book provides qualitative and quantitative methods to analyze and better understand phenomena that change in space and time. An innovative approach is to incorporate ideas and methods from dynamical systems and equivariant bifurcation theory to model, analyze and predict the behavior of mathematical models. In addition, real-life data is incorporated in the derivation of certain models. For instance, the model for a fluxgate magnetometer includes experiments in support of the model.
The book is intended for interdisciplinary scientists in STEM fields, who might be interested in learning the skills to derive a mathematical representation for explaining the evolution of a real system. Overall, the book could be adapted in undergraduate- and postgraduate-level courses, with students from various STEM fields, including: mathematics, physics, engineering and biology.
Author(s): Antonio Palacios
Series: Mathematical Engineering
Publisher: Springer
Year: 2022
Language: English
Pages: 574
City: Cham
Preface
Contents
1 Introduction
1.1 So What Is a Mathematical Model?
1.2 State Variables and Parameters
1.3 Methods and Challenges
1.4 Model Reduction
References
2 Algebraic Models
2.1 Temperature and the Chirping of a Cricket
2.2 Least Squares Fitting of Data
2.2.1 Linear Least Squares Fit
2.2.2 Quadratic Least Squares Fit
2.2.3 General Discrete Least Squares
2.2.4 Cricket Model Revisited
2.2.5 Model Selection
2.2.6 Nonlinear Discrete Least Squares
2.2.7 Gauss-Newton's Method
2.3 The Global Positioning System
2.3.1 Algebraic Equations for GPS Location
2.3.2 Solution via Gauss-Newton's Method
2.3.3 Accuracy
2.4 Allometric Models
2.4.1 Kleiber's Law
2.5 Dimensional Analysis
2.5.1 Free Fall of an Object
2.5.2 Inspection Method
2.5.3 Buckingham Pi Theorem and Method
2.5.4 Allometric Model of Atomic Bomb Blast
2.6 Exercises
References
3 Discrete Models
3.1 Malthusian Growth Model
3.2 Economic Interest Models
3.2.1 Compound Interest
3.2.2 Loans and Amortization
3.3 Time-Dependent Growth Rate
3.3.1 General Population Model
3.3.2 Nonautonomous Malthusian Growth Model
3.3.3 Logistic and Beverton-Holt Models
3.4 Qualitative Analysis of Discrete Models
3.5 Ricker's Model of Salmon Population
3.5.1 Salmon Population in the Skeena River
3.5.2 Analysis of the Ricker's Model
3.6 Heat Exchange
3.7 Newton's Method
3.8 Periodic Points and Bifurcations
3.8.1 Chain Rule and Stability
3.8.2 Period Doubling
3.9 Chaos
3.9.1 Sensitive Dependence
3.9.2 Lyapunov Exponents
3.10 Exercises
References
4 Continuous Models
4.1 Introduction
4.2 Chemostat
4.2.1 Continuous Model of Yeast Growth
4.2.2 Logistic Growth Model
4.3 Qualitative Analysis of Continuous Models
4.3.1 Direction Fields and Phase Portraits in 1D
4.3.2 Stable Manifold Theorem
4.3.3 Phase Portraits
4.4 A Laser Beam Model
4.4.1 Equilibria for Laser
4.4.2 Visualization of Laser Model
4.5 Two Species Competition Model
4.5.1 Qualitative Analysis
4.5.2 Fitting a Competition Model to Yeast Data
4.6 Predator-Prey Model
4.6.1 Sharks and Food Fish
4.6.2 Lotka-Volterra Model
4.7 Method of Averaging
4.7.1 Quasilinear ODE and Lagrange Standard Form
4.8 Linear and Nonlinear Oscillators
4.8.1 Linear Oscillators
4.8.2 Conversion to a System of Differential Equations
4.8.3 Duffing Oscillator
4.8.4 Van der Pol Oscillator
4.8.5 The FitzHugh-Nagumo Model
4.9 Crystal Oscillators
4.9.1 Two-Mode Oscillator Model
4.9.2 Kirchoff's Law of Circuits
4.9.3 Averaging
4.10 Fluxgate Magnetometer
4.10.1 Bistability
4.10.2 What is a Fluxgate Magnetometer
4.10.3 How Does a Fluxgate Magnetometer Work?
4.10.4 Modeling Single-Core Dynamics
4.11 Compartmental Models
4.11.1 General Setting
4.11.2 COVID-19 Modeling
4.12 Exercises
References
5 Bifurcation Theory
5.1 Examples and Phase Portraits
5.2 Conditions for Bifurcations
5.3 Codimension of a Bifurcation
5.4 Codimension One Bifurcations in Discrete Systems
5.4.1 Continuability
5.4.2 Saddle-Node Bifurcation
5.4.3 Transcritical
5.4.4 Pitchfork Bifurcation
5.4.5 Period Doubling Bifurcation
5.4.6 Neimark-Sacker Bifurcation
5.5 Codimension One Bifurcations in Continuous Systems
5.5.1 Saddle-Node Bifurcation
5.5.2 Transcritical Bifurcation
5.5.3 Pitchfork Bifurcation
5.5.4 Nondegeneracy Conditions
5.5.5 Hopf Bifurcation
5.6 Global Bifurcations
5.7 The Role of Symmetry
5.7.1 Continuous Models with Symmetry
5.7.2 Isotropy Subgroups
5.8 Symmetry-Breaking Bifurcations
5.8.1 Steady-State Bifurcations
5.8.2 Equivariant Branching Lemma
5.8.3 Hopf Bifurcation with Symmetry
5.9 Exercises
References
6 Network-Based Modeling
6.1 Routh-Hurwitz Criterion
6.1.1 Spring-Mass System
6.1.2 Stability of Spring-Mass System
6.2 Coupled Cell Systems
6.3 Self-Oscillating Networks
6.4 Unidirectionally Coupled Colpitts Oscillators
6.5 Multifrequency Patterns
6.5.1 Frequency Up-Conversion
6.5.2 Network Configuration
6.5.3 Linear Stability Analysis
6.5.4 The Role of Spatio-Temporal Symmetries
6.5.5 Numerical Simulations
6.5.6 Frequency Down-Conversion
6.5.7 Network Configurations and Symmetries
6.5.8 Simulations
6.6 Feedforward Networks
6.6.1 Hopf Bifurcation
6.6.2 Analysis
6.7 Beam Steering
6.7.1 Array Factor
6.7.2 Signal Amplification
6.8 Coupled Fluxgate System
6.8.1 Network Model
6.8.2 Geometric Description of Solutions by Group Orbits
6.8.3 Onset of Large Amplitude Oscillations
6.8.4 Frequency Response
6.8.5 Sensitivity Response
6.9 Heteroclinic Connections
6.9.1 Finding Heteroclinic Cycles
6.9.2 A Cycle in a Coupled-Cell System
6.10 Exercises
References
7 Delay Models
7.1 Structure and Behavior of Delayed Systems
7.2 System Dynamics with Negative Feedback
7.2.1 Equilibrium Points
7.2.2 Linearization
7.3 Stability Properties
7.4 Epidemic Model
7.5 Lotka-Volterra Model
7.6 Logistic Growth Model with Multiple Delays
7.7 Nyquist Stability Criterion
7.7.1 Transfer Function
7.7.2 Cauchy's Principle of Argument
7.7.3 Examples
7.8 Delay in the Coupled Fluxgate Magnetometer
7.8.1 Model Equations with Multiple Delays
7.8.2 Conversion to Single Delay
7.8.3 Stability Properties of Synchronous Equilibria
7.8.4 Locus of Delay-Induced Oscillations
7.8.5 Generalization to Larger Arrays
7.9 Exercises
References
8 Spatial-Temporal Models
8.1 Reaction-Diffusion Models
8.1.1 Logistic Growth with Diffusion
8.1.2 Heat Equation on Circular Domain
8.1.3 Vibrating Membrane on a Rectangular Domain
8.1.4 Generalization to Higher Dimensions
8.2 Turing Patterns
8.2.1 Diffusion-Driven Instability
8.2.2 Pattern Selection Mechanism
8.3 The Brusselator Model with Diffusion
8.3.1 Linear Stability Analysis
8.3.2 Simulations
8.4 A Model of Flame Instability
8.5 Pattern Formation in Butterflies
8.6 Agent-Based Model of Bubbles in Fluidization
8.6.1 Fluidization Processes
8.6.2 Bubble Dynamics
8.6.3 Computational DIBS Model
8.6.4 Bifurcation Analysis of Single-Bubble Injector
8.6.5 Phase Space Embeddings
8.6.6 Model Fitting
8.6.7 Lyapunov Exponent
8.7 Proper Orthogonal Decomposition
8.7.1 Computational Implementation
8.7.2 The Method of Snapshots
8.8 The Symmetry Perspective
8.8.1 Steady-State Bifurcation in a Triangular Domain
8.8.2 Irreducible Representations
8.8.3 Eigenmodes
8.8.4 Traveling Wave and Standing Wave Patterns
8.9 Exercises
References
9 Stochastic Models
9.1 Definitions
9.2 Stochastic Differential Equations
9.2.1 Itô's Formula
9.2.2 Examples
9.3 Colored Noise
9.3.1 Langevin Equation
9.3.2 Ornstein-Uhlenbeck Process
9.3.3 Euler-Maruyama Numerical Algorithm
9.4 Colored Noise in Bistable Systems
9.5 Fokker-Planck Equation
9.6 Phase Drift in a Network of Gyroscopes
9.6.1 Equations of Motion
9.6.2 Bi-Directionally Coupled Ring
9.6.3 Computational Bifurcation Analysis
9.6.4 Robustness
9.7 Phase Drift in a Model for Precision Timing
9.7.1 Coupled System
9.7.2 Phase Relations
9.7.3 Analysis of Uncoupled Network
9.7.4 Analysis of Unidirectional Coupling
9.7.5 Fundamental Limit
9.8 Stochastic Model of Flame Instability
9.8.1 Computer Simulations
9.8.2 Mode Decomposition
9.8.3 Amplitude Equations
9.9 Exercises
References
10 Model Reduction and Simplification
10.1 Center Manifold Reduction
10.1.1 Computing the Center Manifold
10.1.2 Examples
10.2 Lyapunov-Schmidt Reduction
10.2.1 Computational Aspects
10.2.2 Symmetries
10.2.3 Examples
10.3 Galerkin Projection
10.4 Normal Forms
10.4.1 Hopf Bifurcation
10.4.2 General Method
10.5 Exercises
References
Appendix A MATLAB Programs
A.1 Algebraic Programs
A.2 Discrete Model Programs
A.2.1 Population Models for the United States
A.2.2 Bifurcations in the Discrete Logistic Model
A.2.3 Sensitive Dependence in the Logistic Model
A.3 Continuous Model Programs
A.3.1 Yeast Growth Models
A.3.2 Two Species Competition
A.3.3 Forced Linear Oscillator
A.3.4 Weakly Forced van der Pol Oscillator
A.4 Bifurcation Theory
A.4.1 Sand Dollar Pattern
A.4.2 Neimark-Sacker Bifurcation
A.5 Hybrid Model Programs
A.6 Delay Model Programs
A.6.1 Epidemic Model Programs
A.6.2 Lotka-Volterra Model
A.6.3 Nyquist Plots
A.7 Stochastic Models
A.7.1 Stochastic Model of Stock Prices
A.7.2 Ornstein-Uhlenbeck Process
A.7.3 Fokker-Planck Equation
Appendix B Computations of Phase Drift
Appendix C Proper Orthogonal Decomposition
C.1 Properties
C.2 Consequences of Symmetry
References
Index
