Nonlinear Physics with Maple for Scientists and Engineers

Front Cover
Birkhäuser Boston, 2000 - Computers - 661 pages
Second Updated Edition! Previously published in two separate volumes, this new edition now combines both the standard text and the lab files into one comprehensive volume, and includes a cross-platform CD-ROM containing the Maple code, files and worksheets. 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 second edition extensive use is made of the computer algebra system, Maple V. No prior knowledge of Maple or of programming is assumed. The authors have provided 74 Maple files on a CD-ROM, all classroom tested, as well as 60 annotated Maple worksheets. These files and worksheets may be used to both solve and explore the text's 400 problems. The book includes 30 experimental activities which are intended to deepen and broaden the reader's understanding of the nonlinear physics. These activities are correlated with Part I, the theoretical framework of the text. Reviewer comments on the first edition of Nonlinear Physics with Maple for Scientists and Engineers : 'Correctly balances a good treatment of nonlinear, but also nonchaotic, behavior of systems with some of the exciting findings about chaotic of the book's strengths 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, Maple-based software, and lab manual gives the newcomer to nonlinear physics quite an effective set of tools...this text simultaneously serves as an excellent, structured introduction to Maple V....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 '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 Contents: Preface Part I: THEORY 1. Introduction 1.1 It's a Nonlinear World 1.2 Symbolic Computation 1.2.1 Examples of Maple Operations 1.2.2 Getting Maple Help 1.2.3 Use of Maple in Studying Nonlinear Physics 1.3 Nonlinear Experimental Activities 1.4 Scope of Part I (Theory) 2. Nonlinear Systems, Part I 2.1 Nonlinear Mechanics 2.1.1 The Simple Pendulum 2.1.2 The Eardrum 2.1.3 Nonlinear Damping 2.1.4 Nonlinear Lattice Dynamics 2.2 Competition Phenomena 2.2.1 Volterra-Lotka Competition Equations 2.2.2 Population Dynamics of Fox Rabies in Europe 2.2.3 Eigen and Schuster's Theory of the Selection and Evolution of Biological Molecules 2.2.4 Laser Beam Competition Equations 2.2.5 Rapoport's Model for the Arms Race 2.3 Nonlinear Electrical Phenomena 2.3.1 Nonlinear Inductance 2.3.2 An Electronic Oscillator (the Van der Pol Equation) 2.4 Chemical and Other Oscillators 2.4.1 Chemical Oscillators 2.4.2 The Beating Heart 3. Nonlinear Systems, Part II 3.1 Pattern Formation 3.1.1 Chemical Waves 3.1.2 Snowflakes and Other Fractal Structures 3.1.3 Rayleigh-Benard Convection 3.1.4 Cellular Automata and the Game of Life 3.2 Solitons 3.2.1 Shallow Water Waves (KdV and Other Equations) 3.2.2 Sine-Gordon Equation 3.2.3 Self-Induced Transparency 3.2.4 Optical Solitons 3.2.5 The Jovian Great Red Spot (GRS) 3.2.6 The Davydov Soliton 3.3 Chaos and Maps 3.3.1 Forced Oscillators 3.3.2 Lorenz and Rossler Systems 3.3.3 Poincare Sections and Maps 3.3.4 Examples of One- and Two-Dimensional Maps 4. Topological Analysis 4.1 Introductory Remarks 4.2 Types of Simple Singular Points 4.3 Classifying Simple Singular Points 4.3.1 Poincare's Theorem for the Vortex (Centre) 4.4 Examples of Phase Plane Analysis 4.4.1 The Simple Pendulum 4.4.2 The Laser Competition Equations 4.4.3 Example of a Higher Order Singularity 4.5 Bifurcations 4.6 Isoclines 4.7 3-Dimensional Nonlinear Systems 5. Analytic Methods 5.1 Introductory Remarks 5.2 Some Exact Methods 5.2.1 Separation of Variables 5.2.2 The Bernoulli Equation 5.2.3 The Riccati Equation 5.2.4 Equations of the Structure d 2 y /dx 2 =f(y) 5.3 Some Approximate Methods 5.3.1 Maple Generated Taylor Series Solution 5.3.2 The Perturbation Approach: Poisson's Method 5.3.3 Lindstedt's Method 5.4 The Krylov-Bogoliubov (KB) Method 5.5 Ritz and Galerkin Methods 6. The Numerical Approach 6.1 Finite-Difference Approximations 6.2 Euler and Modified Euler Methods 6.2.1 Euler Method 6.2.2 The Modified Euler Method 6.3 Runge-Kutta (RK) Methods 6.3.1 The Basic Approach 6.3.2 Examples of Common RK Algorithms 6.4 Adaptive Step Size 6.4.1 A Simple Example 6.4.2 The Step Doubling Approach 6.4.3 The RKF 45 Algorithm 6.5 Stiff Equations 6.6 Implicit and Semi-Implicit Schemes 7. Limit Cycles 7.1 Stability Aspects 7.2 Relaxation Oscillations 7.3 Bendixson's First Theorem: The Negative Criterion 7.3.1 Bendixson's Negative Criterion 7.3.2 Proof of Theorem 7.3.3 Applications 7.4 The Poincare-Bendixson Theorem 7.4.1 Poincare-Bendixson Theorem 7.4.2 Application of the Theorem 7.5 The Brusselator Model 7.5.1 Prigogine-LeFever (Brusselator) Model 7.5.2 Application of the Poincare-Bendixson Theorem 7.6 3-Dimensional Limit Cycles 8. Forced Oscillators 8.1 Duffing's Equation 8.1.1 The Harmonic Solution 8.1.2 The Nonlinear Response Curves 8.2 The Jump Phenomenon and Hysteresis 8.3 Subharmonic and Other Periodic Oscillations 8.4 Power Spectrum 8.5 Chaotic Oscillations 8.6 Entrainment and Quasiperiodicity 8.6.1 Entrainment 8.6.2 Quasiperiodicity 8.7 The Rossler and Lorenz Systems 8.7.1 The Rossler Attractor 8.7.2 The Lorenz Attractor 8.8 Hamiltonian Chaos 8.8.1 Hamiltonian Formulation of Classical Mechanics 8.8.2 The Henon-Heiles Hamiltonian 9. Nonlinear Maps 9.1 Introductory Remarks 9.2 The Logistic Map 9.2.1 Introduction 9.2.2 Geometrical Representation 9.3 Fixed Points and Stability 9.4 The Period-Doubling Cascade to Chaos 9.5 Period Doubling in the Real World 9.6 The Lyapunov Exponent 9.7 Stretching and Folding 9.8 The Circle Map 9.9 Chaos versus Noise 9.10 2-Dimensional Maps 9.10.1 Introductory Remarks 9.10.2 Classification of Fixed Points 9.10.3 Delayed Logistic Map 9.10.4 Mandelbrot Map 10. Nonlinear PDE Phenomena 10.1 Introductory Remarks 10.2 Burger's Equation 10.3 Backlund Transformations 10.3.1 The Basic Idea 10.3.2 Examples 10.3.3 Nonlinear Superposition 10.4 Solitary Waves 10.4.1 The Basic Approach 10.4.2 Phase Plane Analysis 10.4.3 KdV Equation 10.4.4 Sine-Gordon Equation 10.4.5 The Three-Wave Problem 11. Numerical Simulation 11.1 Finite Difference Approximations 11.2 Explicit Methods 11.2.1 Diffusion Equation 11.2.2 Fisher's Nonlinear Diffusion Equation 11.2.3 Klein-Gordon Equation 11.2.4 KdV Solitary Wave Collisions 11.3 Von Neumann Stability Analysis 11.3.1 Linear Diffusion Equation 11.3.2 Burger's Equation 11.4 Implicit Methods 11.5 Method of Characteristics 11.5.1 Colliding Laser Beams 11.5.2 General Equation 11.5.3 Sine-Gordon Equation 11.6 Higher Dimensions 12. Inverse Scattering Method 12.1 Lax's Formulation 12.2 Application to KdV Equation 12.2.1 Direct Problem 12.2.2 Time Evolution of the Scattering Data 12.2.3 The Inverse Problem 12.3 Multi-Soliton Solutions 12.4 General Input Shapes 12.5 The Zakharov-Shabat/AKNS Approach Part II: EX

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