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This course will cover dynamical systems theory, and the application of dynamical systems techniques to mathematical, physical, biological, and technological systems described by ordinary differential equations or maps. The primary focus will be on dissipative systems, so that the course is complementary to the Advanced Dynamics sequence (ME 201) which primarily focus on conservative systems.

Igor Mezic

Engineering II 2339

Tel: 893-7603

e-mail: mezic@engineering.ucsb.edu

Lecture Hours: Mon – Wed 11:00-12:15 in Buchanan 1934.

Office Hours: Mon -Wed 12:30-2:00, Engineering II 2339.

Homeworks: Weekly [link].

Office Hours: Mon -Wed 12:30-2:00, Engineering II 2339.

Homeworks: Weekly [link].

- fixed points for vector fields and maps, and their stability properties (Ch. 1, Liapunov functions (Ch. 2)
- invariant manifolds for linear and nonlinear systems (Ch. 3)
- periodic orbits (Ch. 4)
- index theory (Ch. 6)
- asymptotic behavior, attractors (Ch. 8)
- Poincaré-Bendixson Theorem (Ch. 9)
- Poincaré maps (Ch. 10 and 11)
- structural stability (Ch. 12)
- center manifolds (Ch. 18)
- normal forms (Ch. 19)
- bifurcations of fixed points of vector fields (Ch. 20, 22)
- Melnikov’s method (Ch. 28)
- the Smale horseshoe (Ch. 23)
- symbolic dynamics (Ch. 24)
- chaos and strange attractors (Ch. 30).

- J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields”
- S. H. Strogatz, “Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering”
- P. Glendinning, “Stability, Instability, and Chaos”, http://www.scholarpedia.org