Nonlinear Dynamics and Chaotic Phenomena : An Introduction

1 Nonlinear Ordinary Differential Equations -- 1.1 First-order Systems -- 1.1.1 Dynamical System -- 1.1.2 Lipschitz Condition -- 1.1.3 Gronwall’s Lemma -- 1.1.4 Linear Equations -- 1.1.5 Autonomous Equations -- 1.1.6 Stability of Equilibrium Points -- 1.1.6.1 Liapunov and Asymptotic Stability -- 1.1.6.2 Liapunov Function Method -- 1.1.7 Center Manifold Theorem -- 1.2 Phase-plane Analysis -- 1.3 Fully Nonlinear Evolution -- 1.4 Non-autonomous Systems -- 2 Bifurcation Theory -- 2.1 Stability and Bifurcation -- 2.2 Saddle-Node, Transcritical and Pitchfork Bifurcations -- 2.3 Hopf Bifurcation -- 2.4 Break-up of Bifurcations under Perturbations -- 2.5 Bifurcation Theory of One-Dimensional Maps -- 2.6 Appendix: The Normal Form Reduction -- 3 Hamiltonian Dynamics -- 3.1 Hamilton’s Equations -- 3.2 Phase Space -- 3.3 Canonical Transformations -- 3.4 The Hamilton-Jacobi Equation -- 3.5 Action-Angle Variables -- 3.6 Infinitesimal Canonical Transformations -- 3.7 Poisson’s Brackets -- 4 Integrable Systems -- 4.1 Separable Hamiltonian Systems -- 4.2 Integrable Systems -- 4.3 Dynamics on the Tori -- 4.4 Canonical Perturbation Theory -- 4.5 Komogorov-Arnol’d-Moser Theory -- 4.6 Breakdown of Integrability and Criteria for Transition to Chaos -- 4.6.1 Local Criteria -- 4.6.2 Local Stability vs. Global Stability -- 4.6.3 Global Criteria -- 4.7 Magnetic Island Overlap and Stochasticity in Magnetic Confinement Systems -- 4.8 Appendix: The Problem of Internal Resonance in Nonlinearly-Coupled Systems -- 5 Chaos in Conservative Systems -- 5.1 Phasse-Space Dynamics of Conservative Systems -- 5.2 Poincar´e’s Surface of Section -- 5.3 Area-preserving Mappings -- 5.4 Twist Maps -- 5.5 Tangent Maps -- 5.6 Poincar´e-Birkhoff Fixed-Point Theorem -- 5.7 Homoclinic and Heteroclinic Points -- 5.8 Quantitative Measures of Chaos -- 5.8.1 Liapunov Exponents -- 5.8.2 Kolmogorov Entropy -- 5.8.3 Autocorrelation Function -- 5.8.4 Power Spectra -- 5.9 Ergodicity and Mixing -- 5.9.1 Ergodicity -- 5.9.2 Mixing -- 5.9.3 Baker’s Tranformation -- 5.9.4 Lagrangian Chaos in Fluids -- 6 Chaos in Dissipative Systems -- 6.1 Phase-Space Dynamics of Dissipative Systems -- 6.2 Strange Attractors -- 6.3 Fractals -- 6.3.1 Examples of Fractals -- 6.3.2 Box-Counting Method -- 6.4 Multi-fractals -- 6.5 Analysis of Time Series Data -- 6.6 The Lorenz Attractor -- 6.6.1 Equilibrium Solutions and Their Stability -- 6.6.2 Slightly Supercritical Case -- 6.6.3 Existence of an Attractor -- 6.6.4 Chaotic Behavior of the Nonlinear Solutions -- 6.7 Period-Doubling Bifurcations -- 6.7.1 Difference Equations -- 6.7.2 The Logistic Map -- 6.8 Appendix: The Hausdorff-Besicovitch Dimension -- 6.9 Appendix: The Derivation of Lorenz’s Equations -- 6.10 Appendix: The Derivation of Universality for One-Dimensional Maps -- 7 Solitons -- 7.1 Fermi-Pasta-Ulam Recurrence -- 7.2 Korteweg-deVries Equation -- 7.3 Waves in an Anharmonic Lattice -- 7.4 Shallow Water Waves -- 7.5 Ion-acoustic Waves -- 7.6 Basic Properties of Korteweg-deVries Equation -- 7.6.1 Effect of Nonlinearity -- 7.6.2 Effect of Dispersion -- 7.6.3 Similarity Transformation -- 7.6.4 Stokes Waves: Periodic Solutions -- 7.6.5 Solitary Waves -- 7.6.6 Peridic Cnoidal Wave Solutions -- 7.6.7 Interacting Solitary Waves: Hirota’s Method -- 7.7 Inverse-Scattering Transform Method -- 7.7.1 Time Evolution of the Scattering Data -- 7.7.2 Gel’fand-Levitan-Marchenko Equation -- 7.7.3 Direct Scattering Problem -- 7.7.4 Inverse-Scattering Problem -- 7.8 Conservation Laws -- 7.9 Lax Formulation -- 7.10 B¨acklund Transformations -- 8 Singularity Analysis and the Painlev´e Property of Dynamical Systems -- 8.1 The Painlev´e Property -- 8.2 Singularity Analysis -- 8.3 The Painlev´e Property for Partial Differential Equations -- 9 Fractals and Multi-Fractals in Turbulence -- 9.1 Scale Invariance of the Navier-Stokes Equations and the Kolmogorov (1941) Theory -- 9.2 The β -model for Turbulence -- 9.3 The Multi-fractal Models -- 9.4 The Random-β Model -- 9.5 The Transition to Dissipation Range -- 9.6 Critical Phenomena Perspectives on the Turbulence Problem -- 10 Exercises -- 11 References -- 12 Index. - Springer eBooks. - Printed edition: 9789400770935