It's a Nonlinear World

386 Pages·2011·20.245 MB·English

by  Richard H. Enns (auth.)

Springer Undergraduate Texts
in Mathematics and Technology
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Richard H. Enns
It’s a Nonlinear World
1 C
Richard H. Enns
Department of Physics
Simon Fraser University
Burnaby, BC V5A 1S6
Canada
[email protected]

ISSN 1867-5506 e-ISSN 1867-5514
ISBN 978-0-387-75338-6 e-ISBN 978-0-387-75340-9
DOI 10.1007/978-0-387-75340-9
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2010938298
Mathematics Subject Classifi cation (2010): 34A34, 97Mxx
© Springer Science+Business Media, LLC 2011
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v
This text is dedicated
to my loving wife,
Karen,
who lights my path
through
this nonlinear world.
Contents
Preface xi
Part I: WORLD OF MATHEMATICS 1
1 World of Nonlinear Systems 3
1.1 Introduction to Nonlinear ODE Models 4
1.2 Introduction to Difference Equation Models 9
1.3 Solving Nonlinear ODEs on the Computer .. 12
2 World of Nonlinear ODEs 29
2.1 Breakdown of Linear Superposition. 30
2.2 Some Analytically Solvable Examples . 32
2.3 Fixed Points and Phase-Plane Analysis .. 36
2.4 Bifurcations . 44
2.5 Hysteresis and the Jump Phenomena .. 47
2.6 Limit Cycles . 49
2.7 Strange Attractors and Chaos . 53
2.8 Fractal Dimensions . . . 56
2.9 Poincare Sections . 58
2.10 Power Spectrum .. . . 59
3 World of Nonlinear Maps 71
3.1 Fixed Points of One-Dimensional Maps . 71
3.2 Stability Criterion . . . . . . . . . . 73
3.3 Cobweb Diagram. . . . . . . .... 74
3.4 Period Doubling to Chaos . . . . . . 75
3.5 Creating Lorenz Maps. ... . . 77
3.6 Lyapunov Exponent . . . . . . . . . 80
3.7 Two-Dimensional Maps .. . . . . . 82
3.8 Mandelbrot and Julia Sets . 84
3.9 Chaos versus Noise . . . . . . 86
3.10 Controlling Chaos . 89
vii
viii CONTENTS
4 World of Solitons 99
4.1 Korteweg-deVries Solitons .. 101
4.2 Sine-Gordon Solitons . . . . . . . . . . . . . . . . . . 104
4.3 Similarity Solutions. . . . . 107
4.4 Numerical Simulation ... 109
4.4.1 Finite Difference Approximations .. 109
4.4.2 The Zabusky-Kruskal Algorithm . . 111
4.4.3 Method of Characteristics . 114
4.4.4 Numerical Algorithm for the SGE . 115
4.4.5 Numerical Stability. . . . . . . . . 117
4.5 Extension to Cellular Automata . 119
Part II: OUR NONLINEAR WORLD 129
5 World of Motion 131
5.1 Nonlinear Drag or Resistance . 131
5.2 Nonlinear Lift . 137
5.3 The Pendulum, Simple and Otherwise 141
5.3.1 The Simple Pendulum . 141
5.3.2 Parametric Excitation . 146
5.3.3 The Rotating Pendulum . . 147
5.4 Nonlinear Springs . 148
5.4.1 Lattice Dynamics . 152
5.5 Hysteresis and Jumps Revisited . 153
5.6 Precession of Mercury . . . . . . 155
5.7 Saturn's Rings: A "Toy" Model. 158
5.8 Hamiltonian Chaos . 161
6 World of Sports 173
6.1 The Aerodynamics of Sports Balls . . . . . 173
6.2 Bend It Like Beckham ... 176
6.3 A Major League Curveball . 178
6.4 Golf Ball Trajectory . . . . 181
6.5 A Falling Badminton Bird . 184
6.6 Car Racing .... 186
6.7 Medieval Archery . . . . . . . . . . . 187
7 World of Electromagnetism 193
7.1 Nonlinear Electrical Circuits. . . . . . . . . 193
7.1.1 Nonlinear Inductance . 193
7.1.2 Nonlinear Capacitance . . . 196
7.1.3 Chua's Circuit: Piecewise-Linear Negative Resistance 197
7.1.4 Tunnel Diode Oscillator . 199
7.1.5 Josephson Junction . 203
CONTENTS ix
7.1.6 SQUID Magnetometer. . . . . . . . . . . . . . . .... . 208
7.2 Nonlinear Optics . . . . . . . . . .. . . . . . . . . . . .. 209
7.2.1 Optical Soliton Propagation. . . . . . . . . . . . . . . 210
7.2.2 The Navier-Stokes Equations . . . . . . .. 213
7.2.3 Stimulated Scattering of Light 216
7.3 The Earth's Magnetic Field . . . . . . . . . . . . . . . . . . . .. 218
7.3.1 Aurora Borealis. . . . . . . . . . . . . . . . 218
7.3.2 The Drifting North Magnetic Pole . . . . . . . . . . . 221
7.3.3 The Geodynamo Origin of the Earth's Magnetic Field 222
8 World of Weather Prediction 235
8.1 Early History . 236
8.2 The Barotropic Vorticity Equation 237
8.3 Some Meteorological Concepts ... 239
8.4 Modern Numerical Weather Forecasting 244
9 World of Chemistry 255
9.1 Chemical Reactions . 255
9.1.1 Autocatalysis . 255
9.1.2 Michaelis-Menten Enzyme Kinetics .. 257
9.1.3 Lotka-Volterra Mechanism 261
9.2 Chemical Oscillators .. . . 263
9.2.1 The Oregonator . 263
9.2.2 The Brusselator . 266
9.3 Chemical Waves and Patterns . 268
9.3.1 Target Patterns and Spiral Waves . 268
9.3.2 Reaction-Diffusion Equations . 269
9.3.3 How the Leopard Got Its Spots . 272
10 World of Disease 281
10.1 Classifying the Spread of Infectious Diseases .. 282
10.2 Basic Models of Disease Transmission . 283
10.2.1 The SIS Model . 284
10.2.2 The SIR Model without Vital Dynamics . 286
10.2.3 The SIR Model with Vital Dynamics . 289
10.2.4 Herd Immunity and Vaccination .. . . 291
10.2.5 Geographic Spread of an Epidemic . . . . . . . . . . . . . . . . . 292
10.3 Examples of Disease Growth . . 296
10.3.1 Mad Cow Disease. . . . . . . . 296
10.3.2 Avascular Tumor Growth . . . . . . . . . . 300
11 World ofWar 311
11.1 The Coevolutionary Arms Race . 312
11.1.1 The Newt versus the Garter Snake. . . . . . . . . 312
11.1.2 Biological Arms Race with a Dangerous Prey ..... 314
x CONTENTS
11.1.3 The Wild Parsnip and Geographic Mosaic Theory . 322
11.2 Human Conflict . 323
11.2.1 Political Complexity: Nonlinear Models of Politics 323
11.2.2 Richardson Arms Race Model . 324
11.2.3 Chaos-a Model for the Outbreak of War . 326
11.2.4 The Dynamics of Warfare: Lanchester Equations 335
11.2.5 War of the Fire Ants . 337
Bibliography 345
Index 367
Preface
Thepurposeofthistext, It's a Nonlinear World, istopreparescienceandengineering
students for the "real" world where problems and issues on the frontiers of modern
scientific, technological, economic, and social research are often nonlinear in nature. In
this nonlinear world, many ofthe mathematical concepts and tools learned and applied
in traditional undergraduate, and even graduate, science courses are simply inadequate
and new mathematical tools must be introduced. This text will supply these tools and
then illustrate how they are used, drawing examples from diverse fields in the physical,
chemical, biological, engineering, medical, and social sciences.
The book is divided into two parts, the first section introducing the reader to non
linear dynamical (evolving with time) systems in the World of Mathematics. In the
opening chapter of this section, we examine what is meant by a nonlinear mathemat
ical system, providing a variety of historically important, as well as current, examples
formulated in terms ofordinary differentialequations (ODEs) andfinite differenceequa
tions ("maps"). Since exact analytic solutions to nonlinear ODE model equations of
importance in the real world generally do not exist, the reader is introduced to some of
the more common numerical algorithms for solving these equations on the computer.
In the subsequent three chapters ofthe World ofMathematics, wesystematically
present the mathematical framework of nonlinear dynamics. The material is organized
according to mathematical structure, namely, nonlinear ODEs, nonlinear maps, and
similarity and soliton solutions of nonlinear PDE (partial differential equation) models.
In these chapters, the reader is introduced to such nonlinear concepts as fixed points,
bifurcations, limit cycles, fractals, chaos, solitons, etc., and nonlinear diagnostic tools
such as fixed point analysis, bifurcation diagrams, Lyapunov exponents, and so on.
Thesecondpart (Our Nonlinear World) ofthebookpresentsillustrativeexamples
of nonlinear dynamics in the real world grouped in the following seven chapters:
• World of Motion
• World of Sports
• World of Electromagnetism
• World of Weather Prediction
• World of Chemistry
• World of Disease
• World of War
Xl
PREFACE
XII
Each chapter provides topics which are highly relevant to the contemporary world and
makes extensive use of the ideas and methods introduced in the first part of the text.
Such a selection oftopics and examples is inherently bound to be uneven, reflecting not
only the background and knowledge of the author but also the fact that the nonlinear
universeisvast and weare able tosampleonlya smallportionofit. Theexamplesrange
from theflight ofa major baseballleaguecurve ball, totheoriginoftheearth'smagnetic
field, to the spread of an epidemic, to the conflict between different ant colonies, and
to the inherent difficulty in long-range weather forecasting, to mention just a few ofthe
intellectual treats that will be presented. All the examples are fully referenced to the
published literature or the Internet so that they can be more fully explored if desired.
The Internet is becoming a rich source of information with many nonlinear scientists
making copies of their published (and refereed) papers available online. Journals, on
the other hand, do not provide online copies of published papers free of charge, so a
visit to a university or college library may be necessary to view these papers.
The mathematical level of the text assumes a good working knowledge of basic
calculus (ordinary and partial derivatives, integrals, etc.) and a reasonable familiarity
with differential equations. To keep the text as mathematically simple as possible,
the overwhelming number of examples are formulated in terms of nonlinear ODEs and
maps. With the exception of seeking soliton and similarity solutions in certain cases
by reducing PDEs to ODEs, the coverage of most PDE models in this text tends to
generally be more qualitative than quantitative.
It's a Nonlinear World may be used as a course text or for self-study, but is
written in such a way that the more casual mathematically literate reader can simply
read the book for intellectual enjoyment and enlightenment. A wide variety of exercises
and problems are provided at the end of each chapter which allow the reader to explore
other nonlinear models and, if desired, to test his or her mastery of the subject matter.
This book is intended to be open-ended, aimed at whetting the appetite of the
reader to more fully explore our nonlinear world. Entire regions of this world, such as
nonlinear modeling in economics, have not been traversed in this text and remain for
you to discover what treats lie therein.