Nonlinear Dynamics and Chaotic Phenomena: An Introduction

This book starts with a discussion of nonlinear ordinary differential equations, bifurcation theory and Hamiltonian dynamics. It then embarks on a systematic discussion of the traditional topics of modern nonlinear dynamics  -- integrable systems, Poincaré maps, chaos, fractals and strange attractor...

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Main Author: Shivamoggi, Bhimsen K. (Author)
Corporate Author: SpringerLink (Online service)
Format: SpringerLink eBooks
Language:English
Published: Dordrecht : Springer Netherlands : Imprint: Springer, 2014.
Edition:2nd ed. 2014.
Series:Fluid Mechanics and Its Applications, 103
Subjects:
Online Access:http://dx.doi.org/10.1007/978-94-007-7094-2
Table of Contents:
  • 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.