Nonlinear Dynamics

Syllabus

• Surprises in the nonlinear world (Some history and Examples: Logistic equation, Catalysis, Fermi-Pasta-Ulam Problem, Chaotic systems);
• Concepts in dynamical systems (phase space, fixed points, stability, Poincare map);
• Basic theorems in system dynamics (Poincare-Lyapounov, Hartmann-Grobmann, Center Manifold, Review of KAM Theorem);
• Perturbation theory (secular terms, resonance in perturbation theory, Gronwall lemma, error estimation in approximation methods);
• Perturbation method (method of multiple time scales, method of normal forms, freedom in choice of zero-order approximation);
• Applications in ODE's (Duffing oscillator, forced oscillations, limit cycles; Lorentz equations);
• Applications in PDE's (nonlinear diffusion; amplitude equations; nonlinear wave equations - Burgers, KdV & NLS equations and their wave solutions, solitons, compactons);
• Chaos in discrete maps - logistic equation;
• Fractals - basic concepts.

Sources:

1. Arnold, V. I., Geometrical methods in the Theory of Differential equations (Springer-Verlag, NY, 1988).
2. Guckenheimer, J. & Holmes, P. J., Nonlinear Oscillations of Vector Fields (Springer-Verlag, NY, 1998).
3. Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, NY, 1992).
4. Lichtenberg, A. J. & Lieberman, M. A., Regular and Chaotic Dynamics (Springer-Verlag, NY, 1992).
5. Abarbanel, H. D. I., Rabinovich, M. I. & Suschnik, M. M., Introduction to Nonlinear Dynamics for Physicicsts (World Scientific, Singapore, 1993).
6. Nayfeh, A., Perturbation Methods (Wiley, 1978).
7. Hao Bai-Lin, Chaos (World Scientidic, Singapore, 1984).
8. Kahn, P. B., Mathematical Methods for Scientists & Engineers (Wiley, NY, 1990).
9. Kahn, P. B. & Zarmi Y., Nonlinear Dynamics – Exploration Through Normal Forms (Wiley, NY, 1998).