# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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Module/Chapter 12: Chaos and Control

Assumed concepts: The pendulum ODEs of Module 10; systems of ODEs; experience with Poincare sections and/or discrete dynamical systems (Chapter 13) is helpful

The three submodules of this unit tell a story, and in the process illustrate a theorem from current research. This module uses sensitivity to initial conditions and the Poincare section to assist with the analysis. Sinks, saddles, basins, and stability are described. Finally, the elusive boundaries of the Tangled Basin provide a mechanism for control of the chaotically wandering pendulum. The module ends in a fascinating control game that is both fun to play and illuminates the theorem mentioned above.

Module/Chapter 13: Discrete Dynamical Systems

Assumed concepts: Acquaintance with complex numbers and the ideas of equilibrium and stability are helpful

The module provides a gentle introduction to an increasingly important subject. The chapter fills in the technical and mathematical background.

This module could be used successfully in a liberal arts course for students with no calculus.

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Level-of-Difficulty of Modules

The chart below is a handy reference for the levels of the submodules.

Elementary Intermediate Advanced

1

2.1, 2.2 2.3, 2.4

3.1 3.3 3.2

4.1

5.1 5.2 5.3

6.1 6.2 6.3

7.1 7.2

8.1 8.2 8.3 8.4

9.1 9.2 9.3

10.1 10.2

11.1 11.2

12.1

13.1 13.2 13.3

In constructing this chart we have used the following criteria: Elementary: Straightforward, self-contained, can be used as a unit in any introductory calculus or ODE course.

Intermediate: Builds on some prior experience, including earlier submodules and chapters.

Advanced: More challenging models or mathematics, especially suitable for term or group projects.

User's Guide

The User’s Guide is the basic reference for the features of the ODE Architect Tool. The Guide is included on the CD-ROM for ODE Architect.

CONTENTS

1 Modeling with the ODE Architect

1

Douglas Campbell & Wade Ellis, West Valley Community College

Building a Model of the Pacific Sardine Population 2

The Logistic Equation 10

Introducing Harvesting via Landing Data 12

How to Model in Eight Steps 15

Explorations 17

Margie Hale & Michael Branton, Stetson University

Differential Equations 26

Solutions to Differential Equations 26

Solving a Differential Equation 27

Slope Fields 27

Initial Values 28

Finding a Solution Formula 28

Modeling 30

The Juggler 30

The Sky Diver 31

Explorations 35

Margie Hale, Stetson University & Douglas Quinney, University of Keele

Newton's Law of Cooling 44

Cooling an Egg 44

Finding a General Solution 44

Time-Dependent Outside Temperature 46

Air Conditioning a Room 47

The Case of the Melting Snowman 49

Explorations 51

2 Introduction to ODEs

25

3 Some Cool ODEs

43

CONTENTS

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4 Second-Order Linear Equations 57

William Boyce & William Siegmann, Rensselaer Polytechnic Institute

Second-Order ODEs and the Architect 58

Undamped Oscillations 58

The Effect of Damping 60

Forced Oscillations 61

Beats 63

Electrical Oscillations: An Analogy 64 Seismographs 64 Explorations 69

5 Models of Motion 77

Robert Borrelli & Courtney Coleman, Harvey Mudd College Vectors 78

Forces and Newton's Laws 79 Dunk Tank 80

Longer to Rise or to Fall? 81 Indiana Newton 82 Ski Jumping 84 Explorations 85

6 First-Order Linear Systems 93

William Boyce & William Siegmann, Rensselaer Polytechnic Institute Background 94

Examples of Systems: Pizza and Video, Coupled Springs 94 Linear Systems with Constant Coefficients 95 Solution Formulas: Eigenvalues and Eigenvectors 97 Calculating Eigenvalues and Eigenvectors 98 Phase Portraits 99

Using ODE Architect to Find Eigenvalues and Eigenvectors 102 Separatrices 103 Parameter Movies 103 Explorations 105

7 Nonlinear Systems 115

Michael Branton, Stetson University

Linear vs. Nonlinear 116

The Geometry of Nonlinear Systems 116

Linearization 117

Separatrices and Saddle Points 120

Behavior of Solutions Away from Equilibrium Points 121 Bifurcation to a Limit Cycle 122 Higher Dimensions 123

Spinning Bodies: Stability of Steady Rotations 123 The Planar Double Pendulum 126 Explorations 129

8 Compartment Models

Courtney Coleman & Michael Moody, Harvey Mudd College

Lake Pollution 136 Allergy Relief 137 Lead in the Body 139 Equilibrium 141

The Autocatalator and a Hopf Bifurcation 142 Explorations 147

9 Population Models

Michael Moody, Harvey Mudd College

Modeling Population Growth 156 The Logistic Model 156 Two-Species Population Models 158 Predator and Prey 159 Species Competition 160

Mathematical Epidemiology: The SIR Model 161 Explorations 163

10 The Pendulum and Its Friends

John Hubbard & Beverly West, Cornell University

Modeling Pendulum Motion 174 Conservative Systems: Integrals of Motion 176 The Effect of Damping 177 Separatrices 180 Pumping a Swing 182

Writing the Equations of Motion for Pumping a Swing 182 Geodesics 185

Geodesics on a Surface of Revolution 186 Geodesics on a Torus 188 Explorations 193

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