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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: Shivamoggi, Bhimsen K. Nonlinear dynamics and chaotic phenomena. Dordrecht : Springer, [2014] (DLC) 2013955880 (OCoLC)871306999 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Bhimsen K Shivamoggi |

ISBN: | 9789400770942 9400770944 |

OCLC Number: | 880386031 |

Notes: | Previous edition: Dordrecht; London: Kluwer Academic, 1997. |

Description: | 1 online resource (xxvii, 375 pages) : illustrations |

Contents: | Introduction to Chaotic Behavior in Nonlinear Dynamics. Phase-Space Dynamics. Conservative Dynamical Systems. Dissipative Dynamical Systems. Routes to Chaos. Turbulence in Fluids -- 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 Systems. 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 for One-Dimensional Maps. 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. Kolmogorov-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. Appendix. The Problem of Internal Resonances in Nonlinearly-Coupled Systems. 5. Chaos in Conservative Systems. 5.1. Phase-Space Dynamics of Conservative Systems. 5.2. Poincaré's Surface of Section. 5.3. Area-Preserving Mappings. 5.4. Twist Maps. 5.5. Tangent Maps. 5.6. Poincaré-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 Transformation. 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. Appendix A. The Hausdorff-Besicovitch Dimension. Appendix B. The Derivation of Lorenz's Equation. Appendix C. The Derivation of Universality for One-Dimensional Maps. 7. Solitons. 7.1. Fermi-Pasta-Ulam Recurrence. 7.2. Korteweg-de Vries Equation. 7.3. Waves in an Anharmonic Lattice. 7.4. Shallow Water Waves. 7.5. Ion-Acoustic Waves. 7.6. Basic Properties of the Korteweg-de Vries 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. Periodic 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. Inverse Scattering Problem: 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. Bäcklund Transformations. 8. Singularity Analysis and the Painlevé Property of Dynamical Systems. 8.1. The Painlevé Property. 8.2. Singularity Analysis. 8.3. The Painlevé Property for Partial Differential Equations. |

Series Title: | Fluid mechanics and its applications, v. 103. |

Responsibility: | Bhimsen K. Shivamoggi. |

### Abstract:

This book clearly details the basic concepts, language and results of nonlinear dynamical systems. It serves as an ideal reference to first-year graduate students and theoretically inclined researchers in the physical sciences and engineering.
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