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