Nonlinear Dynamics

• 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.

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