Sinossi
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 P
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