Synopsis
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 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 (Center).- 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.- 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 & 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 Pe
Les informations fournies dans la section « Synopsis » peuvent faire référence à une autre édition de ce titre.
