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Elementary differential equations 7th edition - Boyce W.E

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
Download (direct link): elementarydifferentialequat2001.pdf
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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.
xiii
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 Userís 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
xv
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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