Nonlinear Physics With Maple for Scientists and Engineers

by Enns, Richard H.; McGuire, George C.

• ISBN13: 9780817641191
• ISBN10: 081764119X
• Edition: 2nd
• Format: Hardcover
• Copyright: 1999-12-01
• Publisher: Birkhauser

Summary

Extensive use is made of the Maple computer algebra system. The authors have provided 74 Maple files, as well as 60 annotated Maple worksheets, on a CD-ROM. The files and worksheets may be used to solve and explore the text's 400 problems. Includes 30 experimental activities. DLC: Nonlinear theories--Data processing.

Table of Contents

Preface	xi
I THEORY	1	(492)
Introduction	3	(32)
It's a Nonlinear World	3	(3)
Symbolic Computation	6	(25)
Examples of Maple Operations	7	(15)
Getting Maple Help	22	(1)
Use of Maple in Studying Nonlinear Physics	23	(8)
Nonlinear Experimental Activities	31	(1)
Scope of Part I (Theory)	32	(3)
Nonlinear Systems. Part I	35	(42)
Nonlinear Mechanics	35	(15)
The Simple Pendulum	35	(7)
The Eardrum	42	(2)
Nonlinear Damping	44	(3)
Nonlinear Lattice Dynamics	47	(3)
Competition Phenomena	50	(14)
Volterra--Lotka Competition Equations	50	(5)
Population Dynamics of Fox Rabies in Europe	55	(2)
Selection and Evolution of Biological Molecules	57	(3)
Laser Beam Competition Equations	60	(2)
Rapoport's Model for the Arms Race	62	(2)
Nonlinear Electrical Phenomena	64	(7)
Nonlinear Inductance	64	(1)
An Electronic Oscillator (the Van der Pol Equation)	65	(6)
Chemical and Other Oscillators	71	(6)
Chemical Oscillators	71	(4)
The Beating Heart	75	(2)
Nonlinear Systems. Part II	77	(44)
Pattern Formation	77	(16)
Chemical Waves	77	(2)
Snowflakes and Other Fractal Structures	79	(5)
Rayleigh--Benard Convection	84	(2)
Cellular Automata and the Game of Life	86	(7)
Solitons	93	(15)
Shallow Water Waves (KdV and Other Equations)	95	(4)
Sine-Gordon Equation	99	(4)
Self-Induced Transparency	103	(1)
Optical Solitons	104	(3)
The Jovian Great Red Spot (GRS)	107	(1)
The Davydov Soliton	107	(1)
Chaos and Maps	108	(13)
Forced Oscillators	108	(3)
Lorenz and Rossler Systems	111	(2)
Poincare Sections and Maps	113	(3)
Examples of One- and Two-Dimensional Maps	116	(5)
Topological Analysis	121	(42)
Introductory Remarks	121	(4)
Types of Simple Singular Points	125	(4)
Classifying Simple Singular Points	129	(5)
Poincare's Theorem for the Vortex (Center)	134	(1)
Examples of Phase Plane Analysis	134	(16)
The Simple Pendulum	134	(3)
The Laser Competition Equations	137	(6)
Example of a Higher Order Singularity	143	(7)
Bifurcations	150	(3)
Isoclines	153	(2)
3-Dimensional Nonlinear Systems	155	(8)
Analytic Methods	163	(52)
Introductory Remarks	163	(1)
Some Exact Methods	164	(22)
Separation of Variables	166	(4)
The Bernoulli Equation	170	(2)
The Riccati Equation	172	(3)
Equations of the Structure d2y/dx2 = f(y)	175	(11)
Some Approximate Methods	186	(17)
Maple Generated Taylor Series Solution	187	(1)
The Perturbation Approach: Poisson's Method	188	(7)
Lindstedt's Method	195	(8)
The Krylov--Bogoliubov (KB) Method	203	(5)
Ritz and Galerkin Methods	208	(7)
The Numerical Approach	215	(40)
Finite-Difference Approximations	216	(2)
Euler and Modified Euler Methods	218	(11)
Euler Method	219	(4)
The Modified Euler Method	223	(6)
Runge-Kutta (RK) Methods	229	(8)
The Basic Approach	229	(3)
Examples of Common RK Algorithms	232	(5)
Adaptive Step Size	237	(7)
A Simple Example	237	(3)
The Step Doubling Approach	240	(1)
The RKF 45 Algorithm	241	(3)
Stiff Equations	244	(4)
Implicit and Semi-Implicit Schemes	248	(7)
Limit Cycles	255	(28)
Stability Aspects	255	(8)
Relaxation Oscillations	263	(4)
Bendixson's First Theorem	267	(3)
Bendixson's Negative Criterion	267	(1)
Proof of Theorem	267	(2)
Applications	269	(1)
The Poincare--Bendixson Theorem	270	(4)
Poincare--Bendixson Theorem	271	(1)
Application of the Theorem	271	(3)
The Brusselator Model	274	(6)
Prigogine--Lefever (Brusselator) Model	274	(1)
Application of the Poincare--Bendixson Theorem	275	(5)
3-Dimensional Limit Cycles	280	(3)
Forced Oscillators	283	(60)
Duffing's Equation	283	(11)
The Harmonic Solution	286	(2)
The Nonlinear Response Curves	288	(6)
The Jump Phenomenon and Hysteresis	294	(3)
Subharmonic & Other Periodic Oscillations	297	(8)
Power Spectrum	305	(7)
Chaotic Oscillations	312	(12)
Entrainment and Quasiperiodicity	324	(4)
Entrainment	324	(2)
Quasiperiodicity	326	(2)
The Rossler and Lorenz Systems	328	(3)
The Rxsossler Attractor	328	(2)
The Lorenz Attractor	330	(1)
Hamiltonian Chaos	331	(12)
Hamiltonian Formulation of Classical Mechanics	331	(2)
The Henon-Heiles Hamiltonian	333	(10)
Nonlinear Maps	343	(58)
Introductory Remarks	343	(1)
The Logistic Map	344	(7)
Introduction	344	(2)
Geometrical Representation	346	(5)
Fixed Points and Stability	351	(3)
The Period-Doubling Cascade to Chaos	354	(3)
Period Doubling in the Real World	357	(3)
The Lyapunov Exponent	360	(3)
Stretching and Folding	363	(3)
The Circle Map	366	(5)
Chaos versus Noise	371	(5)
2-Dimensional Maps	376	(6)
Introductory Remarks	376	(2)
Classification of Fixed Points	378	(1)
Delayed Logistic Map	379	(1)
Mandelbrot Map	380	(2)
Mandelbrot and Julia Sets	382	(2)
Nonconservative versus Conservative Maps	384	(2)
Controlling Chaos	386	(5)
3-Dimensional Maps: Saturn's Rings	391	(10)
Nonlinear PDE Phenomena	401	(36)
Introductory Remarks	401	(1)
Burgers' Equation	402	(8)
Backlund Transformations	410	(6)
The Basic Idea	410	(1)
Examples	410	(3)
Nonlinear Superposition	413	(3)
Solitary Waves	416	(21)
The Basic Approach	416	(1)
Phase Plane Analysis	417	(4)
KdV Equation	421	(7)
Sine-Gordon Equation	428	(2)
The Three-Wave Problem	430	(7)
Numerical Simulation	437	(36)
Finite Difference Approximations	437	(5)
Explicit Methods	442	(16)
Diffusion Equation	442	(9)
Fisher's Nonlinear Diffusion Equation	451	(1)
Klein--Gordon Equation	452	(3)
KdV Solitary Wave Collisions	455	(3)
Von Neumann Stability Analysis	458	(3)
Linear Diffusion Equation	458	(1)
Burgers' Equation	459	(2)
Implicit Methods	461	(3)
Method of Characteristics	464	(7)
Colliding Laser Beams	464	(3)
General Equation	467	(2)
Sine-Gordon Equation	469	(2)
Higher Dimensions	471	(2)
Inverse Scattering Method	473	(20)
Lax's Formulation	474	(2)
Application to KdV Equation	476	(6)
Direct Problem	476	(3)
Time Evolution of the Scattering Data	479	(2)
The Inverse Problem	481	(1)
Multi-Soliton Solutions	482	(3)
General Input Shapes	485	(2)
The Zakharov--Shabat/AKNS Approach	487	(6)
II EXPERIMENTAL ACTIVITIES	493	(134)
Introduction to Nonlinear Experiments	495	(132)
1 Spin Toy Pendulum	499	(4)
2 Driven Eardrum	503	(4)
3 Nonlinear Damping	507	(4)
4 Anharmonic Potential	511	(6)
5 Iron Core Inductor	517	(4)
6 Nonlinear LRC Circuit	521	(6)
7 Tunnel Diode Negative Resistance Curve	527	(6)
8 Tunnel Diode Self-Excited Oscillator	533	(4)
9 Forced Duffing Equation	537	(6)
10 Focal Point Instability	543	(6)
11 Compound Pendulum	549	(2)
12 Stable Limit Cycle	551	(8)
13 Van der Pol Limit Cycle	559	(4)
14 Relaxation Oscillations: Neon Bulb	563	(6)
15 Relaxation Oscillations: Drinking Bird	569	(4)
16 Relaxation Oscillations: Tunnel Diode	573	(4)
17 Hard Spring	577	(4)
18 Nonlinear Resonance Curve: Mechanical	581	(4)
19 Nonlinear Resonance Curve: Electrical	585	(4)
20 Nonlinear Resonance Curve: Magnetic	589	(4)
21 Subharmonic Response: Period Doubling	593	(2)
22 Diode: Period Doubling	595	(4)
23 Five-Well Magnetic Potential	599	(6)
24 Power Spectrum	605	(4)
25 Entrainment and Quasiperiodicity	609	(2)
26 Quasiperiodicity	611	(2)
27 Chua's Butterfly	613	(4)
28 Route to Chaos	617	(4)
29 Driven Spin Toy	621	(2)
30 Mapping	623	(4)
Bibliography	627	(14)
Index	641