Full Description
Integrability, chaos and patterns are three of the most important concepts in nonlinear dynamics. These are covered in this book from fundamentals to recent developments. The book presents a self-contained treatment of the subject to suit the needs of students, teachers and researchers in physics, mathematics, engineering and applied sciences who wish to gain a broad knowledge of nonlinear dynamics. It describes fundamental concepts, theoretical procedures, experimental and numerical techniques and technological applications of nonlinear dynamics. Numerous examples and problems are included to facilitate the understanding of the concepts and procedures described. In addition to 16 chapters of main material, the book contains 10 appendices which present in-depth mathematical formulations involved in the analysis of various nonlinear systems.
Contents
1. What is Nonlinearity?.- 2. Linear and Nonlinear Oscillators.- 3. Qualitative Features.- 4. Bifurcations and Onset of Chaos in Dissipative Systems.- 5. Chaos in Dissipative Nonlinear Oscillators and Criteria for Chaos.- 6. Chaos in Nonlinear Electronic Circuits.- 7. Chaos in Conservative Systems.- 8. Characterization of Regular and Chaotic Motions.- 9. Further Developments in Chaotic Dynamics.- 10. Finite Dimensional Integrable Nonlinear Dynamical Systems.- 11. Linear and Nonlinear Dispersive Waves.- 12. Korteweg—de Vries Equation and Solitons.- 13. Basic Soliton Theory of KdV Equation.- 14. Other Ubiquitous Soliton Equations.- 15. Spatio-Temporal Patterns.- 16. Nonlinear Dynamics: From Theory to Technology.- A. Elliptic Functions and Solutions of Certain Nonlinear Equations.- Problems.- B. Perturbation and Related Approximation Methods.- B.1 Approximation Methods for Nonlinear Differential Equations.- B.2 Canonical Perturbation Theory for Conservative Systems.- B.2.1 One Degree ol Freedom Hamiltonian Systems.- B.2.2 Two Degrees ol Freedom Systems.- Problems.- C. A Fourth-Order Runge-Kutta Integration Method.- Problems.- Problems.- E. Fractals and Multifractals.- Problems.- Problems.- G. Inverse Scattering Transform for the Schrödinger Spectral Problem.- G.l The Linear Eigenvalue Problem.- G.2 The Direct Scattering Problem.- G.3 The Inverse Scattering Problem.- G.4 Reconstruction of the Potential.- Problems.- H. Inverse Scattering Transform for the Zakharov-Shabat Eigenvalue Problem.- H.1 The Linear Eigenvalue Problem.- H.2 The Direct Scattering Problem.- H.3 Inverse Scattering Problem.- H.4 Reconstruction of the Potentials.- Problems.- I. Integrable Discrete Soliton Systems.- I.1 Integrable Finite Dimensional N-Particles System on a Line: Calogero-Moser System.-I.2 The Toda Lattice.- I.3 Other Discrete Lattice Systems.- I.4 Solitary Wave (Soliton) Solution of the Toda Lattice.- Problems.- J. Painlevé Analysis for Partial Differential Equations.- J.1 The Painlevé Property for PDEs.- J.1.1 Painlevé Analysis.- J.2 Examples.- J.2.1 KdV Equation.- J.2.2 The Nonlinear Schrödinger Equation.- Problems.- References.
