• Instructor: Rafail Abramov
• Office: 1223 SEO
• Office Hours: Friday 1 pm, or after class, but before the department tea starts around 4pm (strictly enforced on the last Friday of each month). Or make an appointment.
• Phone: (312) 413 7945
• E-mail: abramov@math.uic.edu

Textbook: "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields" by J. Guckenheimer and P. Holmes, Springer, 2002, ISBN 0387908196

Additional literature:

• "Ordinary Differential Equations" by V.I. Arnol'd, MIT Press, 1978
• "Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering" by S. Strogatz, Westview Press, 2001
• "Chaotic Evolution and Strange Attractors" by D. Ruelle, Cambridge University Press, 1989
• J.-P. Eckmann, D. Ruelle, "Ergodic Theory of Chaos and Strange Attractors", Rev. Mod. Phys., vol. 57, no. 3, 1985
• "Information Theory and Stochastics for Multiscale Nonlinear Systems" by A.J. Majda, R.V. Abramov and M.J. Grote, AMS, 2005

Tentative syllabus:

Week	Topics
Week 1, Jan 12—16	The Lorenz 63 model, Rossler attractor, evidence for deterministic chaos
Week 2, Jan 21—23	Fixed points, linear dynamics, flows, stable/unstable subspaces
Week 3, Jan 26—30	Nonlinear dynamics, tangent map, stable/unstable manifolds, Lyapunov exponents
Week 4, Feb 2—6	Numerical methods for computing Lyapunov exponents and stable/unstable subspaces. Lyapunov exponents for Lorenz 63 model. Numerical simulations with Lorenz 63 model.
Week 5, Feb 9—13	Statistical description of dynamical systems, invariant probability states, ergodicity. Statistical quantities, entropy. The Lorenz 96 model.
Week 6, Feb 16—20	The Lorenz 96 model (cont'd), bands of linearly stable and unstable waves, phase/group velocities, relevance to atmosphere dynamics, statistical properties, Lyapunov exponents for different dynamical regimes. Linear response to small external perturbations.
Week 7, Feb 23—27	The linear response formula (cont'd), computational methods. The T21 baroptropic truncation on a sphere.
Week 8, Mar 2—6	T21 barotropic truncation (cont'd), its linear response, applications to climate change. Linear response formula for hyperbolic systems, structural stability.
Week 9, Mar 9—13	Statistical description for conservative systems. Examples of energy-conserving systems. Liouville equation, Liouville property.
Week 10, Mar 16—20	Uniform distribution on a sphere of constant energy. Statistical theory for the conservative Lorenz 96 system and the truncated Burgers-Hopf system.
Week 11, Mar 30 — Apr 3	Statistical theory for the standard truncation of the barotropic flow on a torus.
Week 12, Apr 6—10	Classical fluctuation-dissipation theorem for conservative systems.
Week 13, Apr 13—17	Dynamical systems with additional conserved quantities.
Week 14, Apr 20—25	Classical fluctuation-dissipation theorem for forced-dissipative systems. Blended response algorithm.
Week 15, Apr 27 — May 1	Classical fluctuation-dissipation theorem for stochastically driven systems. (If time permits)

Lecture notes: I am writing a set of lecture notes as the class progresses.

Homework:
Homework 1, due Friday Jan 30.
Homework 2, due Friday Feb 20.

I will be continuously updating this webpage as the course progresses. Please watch this webpage, although I will try to announce any further changes by e-mail.