**by J.M.A. Danby, 422 Pages,
Softbound, 8.5 by 11 inches,
229 Illustrations, Includes CD-ROM, w
ith IBM-PC Software
$34.95.
**

**About Computer Modeling: From Sports to Spacefilght . . . From Order to Chaos**

Calculus is the science of change and differential equations are the engines of calculus. A differential equation is like a machine that is capable of movement and ready for action. It only requires some activation: pressing a button, inserting a coin, lighting a fuse — or running a computer. Then it will describe and give details of something that is continually changing. A chemical compound is formed or broken up (gradually, one hopes), a mortgage is repaid, a population increases, a disease spreads, a miss-hit golf ball curves away from its intended direction or an orbit is described in space: in each of these cases the changes will be modeled by differential equations.

Much of the material in this book has been used for over fifteen years in teaching an interdisciplinary one semester course in computer modeling to undergraduates, principally in engineering, physics and mathematics. The book is structure for those who do their own programming, but is also suitable for individuals using "packages" for solving systems of differential equations. The subjects covered are exceptionally diverse and include chaotic systems; population growth and ecology; sickness and health; competition and economics; sports; travel and recreation; space travel and astronomy; pendulums; springs; chemical and other reacting systems.

Accompanying each book is software on CD-ROM which includes over 50 projects from this book — denoted in the text by a CD-ROM symbol. The programs are menu driven with ample help files and make use of graphic and animation techniques to demonstrate the various phenomena being modeled. The programs are written to be run on MS-DOS platforms. The minimum hardware configuration is an IBM compatible 386-level machine with math coprocessor, mouse and VGA color monitor. The programs require 8.5 Mb of disk space and consist of executable files. Pascal code is also included for the user's information.

**About the author:**

J.M.A. Danby was born in 1929, and studied mathematics at Oxford and Manchester Universities. He has taught in the U.S. since 1957, before which he was for a time, a professional musician, as first chair oboist in the London Philharmonic Orchestra. He has taught in the Astronomy Departments of the University of Minnesota and Yale University. For more that 30 years he has been a member of the Mathematics Department at North Carolina State University where he was appointed Alumni Distinguished Professor in recognition of teaching. The minor planet *Danby* was named after him in recognition of his contributions to the study of Celestial Mechanics by the International Astronomical Union. He authored *Fundamentals of Celestial Mechanic*s and co-authored *Astrophysics Simulations* which included software that received an Annual Educational Software award from Computers in Physics in 1996.

**Table of Contents**

**Chapter 1 Getting to Know a Differential Equation**

1.1 What a Differential Equation Might Mean

1.2 What is a Differential Equation Telling Us To Do?

1.3 Looking at and Interpreting a Solution

1.4 When is there a solution? Or when is there a solution? Or when is there a solution?

1.5 Assignments

**Chapter 2 Some Fundamental Concepts in the Solution of Differential Equations**

2.1 Euler's Method

2.2 Truncation Error

2.3 The Order of a Method

2.4 Numerical Confirmation of the Order of Euler's Method

**Chapter 3 One Approach to Solving a System of Ordinary Differential Equations**

3.1 Generalities

3.2 A Look at the Improved Euler's Method

3.3 Runge-Kutta Formulas

3.4 Stepsize Control—Some Tactics to Avoid

3.5 Fehlberg's RKF4(5) Method

3.6 The Implementation of RKF4(5). One Equation

3.7 The Implementation of RKF4(5). A System of Equations

3.8 Projects for Debugging, Testing and Understanding the Program

3.9 Running the Program

**Chapter 4 Introduction to Modeling**

4.1 Generalities

4.2 A Model for Discussion: Richardson's "Arms Race"

4.3 Linear Systems, Linear Stability and Nonlinear Stability

**Chapter 5 An Introduction to Chaotic Systems**

5.1 Introduction

5.2 Chaos in Dynamical Systems

5.3 The Lorenz Equations

5.4 A Periodically Forced Pendulum

5.5 Chaos in Difference Equations

**Chapter 6 Population Growth and Ecology**

6.1 The Predator-Prey Model of Volterra

6.2 Volterra's Model with Periodic Birth-rate

6.3 The Predator-Prey Model with Fishing

6.4 The Predator-Prey Model with Logistic Growth for the Prey

6.5 The Predator-Prey Model with Logistic Growth for Both Species

6.6 An Alternative Law for Predation

6.7 A Predator-Prey Model Due to R.M.May: Limit Cycles

6.8 The Predator-Prey Model with Internal Competition

6.9 Cooperation Between Two Species

6.10 Competition Between Two Species

6.11 The Predator-Prey Model with Child Care

6.12 Predator-Prey Models with More than Two Species

6.13 Carnivores, Vegetarians and Plants

6.14 Violets, Ants and Rodents

6.15 A Model for the Population Growth of a Parasite

6.16 A Model for Cannibalism

6.17 Population Growth in a Changing Environment

6.18 A Model for Lake Pollution

6.19 A Model in Ecology: The Spruce Budworm versus the Balsam Fir

**Chapter 7 Sickness and Health**

7.1 A Model for the Spread of Disease

7.2 Possible Effects of Vaccination

7.3 Possible Effects of Migration

7.4 The Spread of Disease in a Population with Birth and Death Included in the Model: 1

7.5 The Spread of Disease in a Population with Birth and Death Included in the Model: 2

7.6 Cross-Infection Between Two Species

7.7 The Spread of Disease with Incubation Included

7.8 Seasonal Changes in Infectiousness

7.9 The Epidemiology of Malaria

7.10 The Spread of Gonorrhea

7.11 A Model for the Initial Spread of the HIV Virus

7.12 A Model for Weight Change

7.13 Zeeman's Model for the Heartbeat

**Chapter 8 Competition and Economics**

8.1 Lanchester's Combat Models

8.2 Production and Exchange

8.3 The Economics of Fishing: One Species of Fish

8.4 The Economics of Fishing: Two Species of Fish

8.5 Goodwin's Growth Cycle

8.6 A One-sector, Two-capital Model of Economic Growth

**Chapter 9 Sports**

9.1 The Dynamics of a Spinning Ball

9.2 A Model for a Baseball Pitch

9.3 Pitching a Knuckle Ball

9.4 The Flight of a Fly Ball

9.5 Pitching a Softball

9.6 Driving a Golf Ball

9.7 Serving in Tennis

9.8 Kicking a Ball in Soccer

9.9 Bowling a Cricket Ball

9.10 The Swing of a Cricket Ball

9.11 Shuttlecock Trajectories in Badminton

9.12 Table Tennis

9.13 Shooting in Basketball

9.14 A Badly Kicked Football

9.15 The Path of a Discus

9.16 The Motion of a Javelin

9.17 A Model for the Ski Jump

9.18 Running

9.19 Diving

9.20 The Pole Vault

**Chapter 10 Travel and Recreation**

10.1 The Dynamics of Flight

10.2 The Motion of a Hovercraft

10.3 Pitching and Rolling at Sea

10.4 The Motion of a Balloon and its Payload

10.5 Jogging with a Companion

10.6 A Bunjy Jump

10.7 A Model for the Motion of a Yo-yo

10.8 Chaos in the Amusement Park

10.9 Playing Ball in a Space Station

10.10 Fireworks

**Chapter 11 Space Travel and Astronomy**

11.1 The Motion of Three Bodies

11.2 A Trip to the Moon

11.3 The Motion of a Space Station Around L_{4} or L_{5}: 1

11.4 The Motion of a Space Station Around L_{4} or L_{5}: 2

11.5 The Descent of Skylab

11.6 The Range of an ICBM

11.7 The Accuracy of an ICBM

11.8 Aero-Braking the Orbit of a Spacecraft

11.9 The Motion of a Rocket. 1: Introduction

11.10 The Motion of a Rocket. 2: Multi-Stage Rockets

11.11 The Motion of a Rocket. 3: Two-Dimensional Motion

11.12 The Motion of a Rocket. 4: Low-Thrust Orbits

11.13 Reentering the Earth's Atmosphere

11.14 De-spinning a Satellite 1: A Space Yo-yo

11.15 De-spinning a Satellite 2: The Stretch Yo-yo

11.16 If You Were Jupiter, Could You Catch a Comet?

11.17 Using Jupiter to Boost the Orbit of a Spacecraft

11.18 A Grand Tour of the Solar System

11.19 The Motion of the Perihelion of a Planetary Orbit under General Relativity: An Application of the Method of Variation of Parameters

11.20 The Poynting-Robertson Effect: or "How would you like to be a grain of dust in the Solar System?"

11.21 Gravitational Interaction Between Two Galaxies

11.22 If You Were on Venus, Could You See the Back of Your Head?

11.23 Helium Burning in a Hot Star

11.24 An Approximate Model for a Star

11.25 The Limiting Mass of a White Dwarf

**Chapter 12 Pendulums**

12.1 The Simple Pendulum

12.2 The Period of a Simple Pendulum

12.3 The Pendulum with Linear Damping

12.4 The Pendulum with Dry Friction

12.5 The Pendulum of a Clock

12.6 The Simple Pendulum in a Stiff Wind

12.7 The Simple Pendulum With Added Constant Load

12.8 Pull-out Torques of Synchronous Motors

12.9 A Gravity Pendulum

12.10 A Magnetic Pendulum in Two Dimensions

12.11 A Child on a Swing: 1

12.12 A Child on a Swing: 2

12.13 A Pendulum with Varying Length

12.14 A Swinging Censer

12.15 A Pendulum with Moving Pivot

12.16 A Pendulum Wrapped Around a Peg

12.17 The Motion of a Sliding Pendulum: 1

12.18 The Motion of a Sliding Pendulum: 2

12.19 The Swinging Atwood Machine

12.20 The Motion of a Pendulum Connected to a Mass Moving on a Table

12.21 A Double Pendulum

12.22 The Motion of a Pendulum Attached to a Freely Spinning Wheel

12.23 A Magnetic Pendulum in Three Dimensions

12.24 A Dumbbell Satellite

12.25 The Rotation of the Moon

12.26 The Rotation of Hyperion

12.27 The Rotation of Mercury

12.28 The Rotation of the Moon and Mercury, Including Effects of Tidal Friction

**Chapter 13 Springs**

13.1 Non-Linear Springs

13.2 Duffing's Equation

13.3 Van der Pol's Equation, and Friends

13.4 Variation of Parameters and Van der Pol's Equation

13.5 A Dynamic Model for the Buckling of a Column

13.6 Zeeman's Catastrophe Machine

13.7 Two Attracting Wires

13.8 The Action Between a Violin Bow and a String

13.9 Landing an Airplane on an Aircraft Carrier

13.10 A Pendulum with Sprung Pivot

13.11 A Spring Pendulum

13.12 A Chaotic Driven Wheel

**Chapter 14 Chemical and other Reacting Systems**

14.1 The Decomposition of a Molecule

14.2 Enzyme Kinetics

14.3 An Application of Enzyme Kinetics

14.4 More Enzyme Kinetics

14.5 Still More Enzyme Kinetics

14.6 Limit Cycles in Chemical Reactions: The "Brusselator"

14.7 Limit Cycles in Chemical Reactions: The "Oregonator"

14.8 Chemical-Tank-Reactor Stability

14.9 Temperature and Volume Control in a Tank

14.10 The Dynamics of a Reservoir System

**Chapter 15 Bits and Pieces**

15.1 Fireflies

15.2 Curves of Pursuit

15.3 Low-level Bombing

15.4 A Carbon Microphone Circuit

15.5 The Motion of a Ball in a Rotating Circular Ring: 1

15.6 The Motion of a Ball in a Rotating Circular Ring: 2

15.7 A Compass Needle in an Oscillating Magnetic Field

15.8 The Motion of a Piston and Flywheel

15.9 Watt's Governor

15.10 A Dynamo System with Magnetic Reversal

15.11 A Two-Magnet Toy

15.12 Bernoulli's Problem

**Appendix A Lagrange's Equations**

A.1 Introduction

A.2 The Simple Pendulum

**Appendix B Software**

B.1 Introduction

B.2 Chaotic Systems

B.3 Predator-Prey Models

B.4 Sickness and Health

B.5 Sports

B.6 Spaceflight and Astronomy

B.7 Pendulums

B.8 Bits and Pieces

**Bibliography**

**Index**