Nonlinear Physics With Mathematica for Scientists and Engineers

by Enns, Richard H.; McGuire, George

• ISBN13: 9780817642235
• ISBN10: 0817642234
• Edition: CD
• Format: Hardcover
• Copyright: 2001-06-01
• Publisher: Birkhauser

Summary

Nonlinear physics continues to be an area of dynamic modern research, with applications to physics, engineering, chemistry, mathematics, computer science, biology, medicine and economics. In this text extensive use is made of the Mathematica computer algebra system. No prior knowledge of Mathematica or programming is assumed. The authors have included a CD-ROM that contains over 130 annotated Mathematica files. These files may be used to solve and explore the text's 400 problems. This book includes 33 experimental activities that are designed to deepen and broaden the reader's understanding of nonlinear physics. These activities are correlated with Part I, the theoretical framework of the text.Additional features:* User-friendly, accessible presentation integrating theory, experiments, and the provided Mathematica notebooks; as the concepts of nonlinear science are developed, readers are gently introduced to Mathematica as an auxiliary tool* CD-ROM includes a wide variety of illustrative nonlinear examples solved with Mathematica--command structures introduced on a need-to-know basis* Notebooks designed to make use of Mathematica's sound capability* Mathematica notebook using the EulerEquation command incorporated into the textThis work is an excellent text for undergraduate and graduate students as well as a useful resource for working scientists.Reviewer comments on the Maple edition of NONLINEAR PHYSICS:"An...excellent book...the authors have been able to cover an extraordinary range of topics and hopefully excite a wide audience to investigate nonlinear phenomena...accessible to advanced undergraduates and yet challenging enough for graduate students and working scientists.... The reader is guided through it all with sound advice and humor.... I hope that many will adopt the text."-American Journal of Physics"Its organization of subject matter, clarity of writing, and smooth integration of analytic and computational techniques put it among the very best...Richard Enns and George McGuire have written an excellent text for introductory nonlinear physics."-Computers in Physics"...correctly balances a good treatment of nonlinear, but also nonchaotic, behavior of systems with some of the exciting findings about chaotic dynamics...one of the book's strength is the diverse selection of examples from mechanical, chemical, electronic, fluid and many other systems .... Another strength of the book is the diversity of approaches that the student is encouraged to take...the authors have chosen well, and the trio of text,...software, and lab manual gives the newcomer to nonlinear physics quite an effective set of tools.... Basic ideas are explained clearly and illustrated with many examples."-Physics Today"... the care that the authors have taken to ensure that their text is as comprehensive, versatile, interactive, and student-friendly as possible place this book far above the average."-Scientific Computing World

Table of Contents

Preface	xi
I THEORY	1	(510)
Introduction	3	(36)
It's a Nonlinear World	3	(3)
Symbolic Computation	6	(29)
Examples of Mathematica Operations	7	(19)
Getting Mathematica Help	26	(1)
Use of Mathematica in Studying Nonlinear Physics	27	(8)
Nonlinear Experimental Activities	35	(2)
Scope of Part I (Theory)	37	(2)
Nonlinear Systems. Part I	39	(42)
Nonlinear Mechanics	39	(14)
The Simple Pendulum	39	(7)
The Eardrum	46	(2)
Nonlinear Damping	48	(3)
Nonlinear Lattice Dynamics	51	(2)
Competition Phenomena	53	(15)
Volterra-Lotka Competition Equations	54	(5)
Population Dynamics of Fox Rabies in Europe	59	(3)
Selection and Evolution of Biological Molecules	62	(2)
Laser Beam Competition Equations	64	(2)
Rapoport's Model for the Arms Race	66	(2)
Nonlinear Electrical Phenomena	68	(8)
Nonlinear Inductance	68	(1)
An Electronic Oscillator (the Van der Pol Equation)	69	(7)
Chemical and Other Oscillators	76	(5)
Chemical Oscillators	76	(4)
The Beating Heart	80	(1)
Nonlinear Systems. Part II	81	(44)
Pattern Formation	81	(15)
Chemical Waves	81	(2)
Snowflakes and Other Fractal Structures	83	(6)
Rayleigh-Benard Convection	89	(1)
Cellular Automata and the Game of Life	90	(6)
Solitons	96	(15)
Shallow Water Waves (KdV and Other Equations)	99	(4)
Sine-Gordon Equation	103	(4)
Self-Induced Transparency	107	(1)
Optical Solitons	107	(3)
The Jovian Great Red Spot (GRS)	110	(1)
The Davydov Soliton	111	(1)
Chaos and Maps	111	(14)
Forced Oscillators	112	(3)
Lorenz and Rossler Systems	115	(2)
Poincare Sections and Maps	117	(2)
Examples of One- and Two-Dimensional Maps	119	(6)
Topological Analysis	125	(42)
Introductory Remarks	125	(4)
Types of Simple Singular Points	129	(4)
Classifying Simple Singular Points	133	(5)
Poincare's Theorem for the Vortex (Center)	137	(1)
Examples of Phase Plane Analysis	138	(16)
The Simple Pendulum	138	(3)
The Laser Competition Equations	141	(6)
Example of a Higher Order Singularity	147	(7)
Bifurcations	154	(3)
Isoclines	157	(2)
3-Dimensional Nonlinear Systems	159	(8)
Analytic Methods	167	(56)
Introductory Remarks	167	(1)
Some Exact Methods	168	(25)
Separation of Variables	171	(4)
The Bernoulli Equation	175	(2)
The Riccati Equation	177	(2)
Equations of the Structure d2y/dx2 = f(y)	179	(14)
Some Approximate Methods	193	(17)
Mathematica Generated Taylor Series Solution	193	(3)
The Perturbation Approach: Poisson's Method	196	(7)
Lindstedt's Method	203	(7)
The Krylov-Bogoliubov (KB) Method	210	(6)
Ritz and Galerkin Methods	216	(7)
The Numerical Approach	223	(42)
Finite-Difference Approximations	224	(3)
Euler and Modified Euler Methods	227	(11)
Euler Method	228	(4)
The Modified Euler Method	232	(6)
Runge-Kutta (RK) Methods	238	(9)
The Basic Approach	238	(3)
Examples of Common RK Algorithms	241	(6)
Adaptive Step Size	247	(6)
A Simple Example	247	(2)
The Step Doubling Approach	249	(1)
The RKF 45 Algorithm	250	(3)
Stiff Equations	253	(5)
Implicit and Semi-Implicit Schemes	258	(5)
Some Remarks on NDSolve	263	(2)
Limit Cycles	265	(28)
Stability Aspects	265	(8)
Relaxation Oscillations	273	(5)
Bendixson's First Theorem	278	(3)
Bendixson's Negative Criterion	278	(1)
Proof of Theorem	278	(2)
Applications	280	(1)
The Poincare-Bendixson Theorem	281	(4)
Poincare-Bendixson Theorem	282	(1)
Application of the Theorem	282	(3)
The Brusselator Model	285	(6)
Prigogine-Lefever (Brusselator) Model	285	(1)
Application of the Poincare-Bendixson Theorem	286	(5)
3-Dimensional Limit Cycles	291	(2)
Forced Oscillators	293	(62)
Duffing's Equation	293	(11)
The Harmonic Solution	296	(2)
The Nonlinear Response Curves	298	(6)
The Jump Phenomenon and Hysteresis	304	(4)
Subharmonic & Other Periodic Oscillations	308	(8)
Power Spectrum	316	(8)
Chaotic Oscillations	324	(11)
Entrainment and Quasiperiodicity	335	(4)
Entrainment	335	(2)
Quasiperiodicity	337	(2)
The Rossler and Lorenz Systems	339	(3)
The Rossler Attractor	339	(1)
The Lorenz Attractor	340	(2)
Hamiltonian Chaos	342	(13)
Hamiltonian Formulation of Classical Mechanics	342	(1)
The Henon-Heiles Hamiltonian	343	(12)
Nonlinear Maps	355	(58)
Introductory Remarks	355	(1)
The Logistic Map	356	(7)
Introduction	356	(2)
Geometrical Representation	358	(5)
Fixed Points and Stability	363	(3)
The Period-Doubling Cascade to Chaos	366	(3)
Period Doubling in the Real World	369	(3)
The Lyapunov Exponent	372	(3)
Stretching and Folding	375	(3)
The Circle Map	378	(5)
Chaos versus Noise	383	(5)
2-Dimensional Maps	388	(6)
Introductory Remarks	388	(2)
Classification of Fixed Points	390	(1)
Delayed Logistic Map	391	(1)
Mandelbrot Map	392	(2)
Mandelbrot and Julia Sets	394	(2)
Nonconservative versus Conservative Maps	396	(2)
Controlling Chaos	398	(6)
3-Dimensional Maps: Saturn's Rings	404	(9)
Nonlinear PDE Phenomena	413	(38)
Introductory Remarks	413	(1)
Burgers' Equation	414	(11)
Backlund Transformations	425	(7)
The Basic Idea	425	(1)
Examples	425	(4)
Nonlinear Superposition	429	(3)
Solitary Waves	432	(19)
The Basic Approach	432	(1)
Phase Plane Analysis	433	(4)
KdV Equation	437	(5)
Sine-Gordon Equation	442	(3)
The Three-Wave Problem	445	(6)
Numerical Simulation	451	(40)
Finite Difference Approximations	451	(6)
Explicit Methods	457	(16)
Diffusion Equation	457	(9)
Fisher's Nonlinear Diffusion Equation	466	(1)
Klein-Gordon Equation	467	(3)
KdV Solitary Wave Collisions	470	(3)
Von Neumann Stability Analysis	473	(3)
Linear Diffusion Equation	473	(1)
Burgers' Equation	474	(2)
Implicit Methods	476	(3)
Method of Characteristics	479	(7)
Colliding Laser Beams	479	(3)
General Equation	482	(2)
Sine-Gordon Equation	484	(2)
Higher Dimensions	486	(5)
2-Dimensional Reaction-Diffusion Equations	487	(1)
3-Dimensional Light Bullet Collisions	488	(3)
Inverse Scattering Method	491	(20)
Lax's Formulation	492	(3)
Application to KdV Equation	495	(6)
Direct Problem	495	(2)
Time Evolution of the Scattering Data	497	(3)
The Inverse Problem	500	(1)
Multi-Soliton Solutions	501	(2)
General Input Shapes	503	(2)
The Zakharov-Shabat/AKNS Approach	505	(6)
II EXPERIMENTAL ACTIVITIES	511	(144)
Introduction to Nonlinear Experiments	513	(4)
Magnetic Force	517	(4)
Magnetic Tower	521	(4)
Spin Toy Pendulum	525	(4)
Driven Eardrum	529	(4)
Nonlinear Damping	533	(4)
Anharmonic Potential	537	(6)
Iron Core Inductor	543	(4)
Nonlinear LRC Circuit	547	(6)
Tunnel Diode Negative Resistance Curve	553	(6)
Tunnel Diode Self-Excited Oscillator	559	(4)
Forced Duffing Equation	563	(6)
Focal Point Instability	569	(6)
Compound Pendulum	575	(2)
Damped Simple Pendulum	577	(2)
Stable Limit Cycle	579	(8)
Van der Pol Limit Cycle	587	(4)
Relaxation Oscillations: Neon Bulb	591	(6)
Relaxation Oscillations: Drinking Bird	597	(4)
Relaxation Oscillations: Tunnel Diode	601	(4)
Hard Spring	605	(4)
Nonlinear Resonance Curve: Mechanical	609	(4)
Nonlinear Resonance Curve: Electrical	613	(4)
Nonlinear Resonance Curve: Magnetic	617	(4)
Subharmonic Response: Period Doubling	621	(2)
Diode: Period Doubling	623	(4)
Five-Well Magnetic Potential	627	(6)
Power Spectrum	633	(4)
Entrainment and Quasiperiodicity	637	(2)
Quasiperiodicity	639	(2)
Chua's Butterfly	641	(4)
Route to Chaos	645	(4)
Driven Spin Toy	649	(2)
Mapping	651	(4)
Bibliography	655	(14)
Index	669