Nonlinear physics with Maple for scientists and engineers Book

PrefacePart 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-Bénard 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 Rössler Systems 3.3.3 Poincaré 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 Poincaré'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 d2y/dx2=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 Rungé-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 Poincaré-Bendixson Theorem 7.4.1 Poincaré-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 Poincaré-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 Rössler and Lorenz Systems 8.7.1 The Rössler Attractor 8.7.2 The Lorenz Attractor 8.8 Hamiltonian Chaos 8.8.1 Hamiltonian Formulation of Classical Mechanics 8.8.2 The Hénon-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 Bäcklund 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: EXPERIMENTAL ACTIVITIES Introduction to Nonlinear Experiments 1. Spin Toy Pendulum 2. Driven Eardrum 3. Nonlinear Damping 4. Anaharmonic Potential 5. Iron Core Conductor 6. Nonlinear LRC Circuit 7. Tunnel Diode Negative Resistance Curve 8. Tunnel Diode Self-Excited Oscillator 9. Forced Duffing Equation 10. Focal Point Instability 11. Compound Pendulum 12. Stable Limit Cycle 13. Van der Pol Limit Cycle 14. Relaxation Oscillations: Neon Bulb 15. Relaxation Oscillations: Drinking Bird 16. Relaxation Oscillations: Tunnel Diode 17. Hard Spring 18. Nonlinear Resonance Curve: Mechanical 19. Nonlinear Resonance Curve: Electrical 20. Nonlinear Resonance Curve: Magnetic 21. Subharmonic Response: Period Doubling 22. Diode: Period Doubling 23. Five-Well Magnetic Potential 24. Power Spectrum 25. Entrainment and Quasiperiodicity 26. Quasiperiodicity 27. Chua's Butterfly 28. Route to Chaos 29. Driven Spin Toy 30. Mapping Index