calendar:
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Su Mo Tu We Th Fr Sa week no and events
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30 31 1 2 3 4 5 1 2. february start of lectures
6 7 8 9 10 11 12 2
13 14 15 16 17 18 19 3
20 21 22 23 24 25 26 4 21. february presidents day
27 28 1 2 3 4 5 5 march
6 7 8 9 10 11 12 6
13 14 15 16 17 18 19 7
20 21 22 23 24 25 26 8
27 28 29 30 31 1 2 april spring break
3 4 5 6 7 8 9 9
10 11 12 13 14 15 16 10
17 18 19 20 21 22 23 11
24 25 26 27 28 29 30 12
1 2 3 4 5 6 7 13 may end of classes
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8 9 10 11 12 13 14 reading period
15 16 17 18 19 20 21 19. start of exam period
22 23 24 25 26 27 28
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tentative syllabus:
1. Week Introduction:
Wed: Overview and organization of the course.
What is a dynamical system?
Fri: Examples of dynamical systems
2. Week Feigenbaum: maps in one dimensions
Mon: Maps on the interval
Periodic points and their stability.
Wed: Bifurcation of periodic points
Invariant measures
Fri: The dynamical zeta function
The Lyapunov exponent
3. Week Henon: maps in two dimensions
Mon: Examples.
Periodic points and their nature
Wed: Stable manifold theorem and homoclinic points
Construction of stable manifolds
Fri: Lyapunov exponents and random matrices
Definitions of chaos
4. Week Hilbert: ODEs in two dimensions
Wed: Differential equations in the plane and torus
Poincare-Bendixon
Limit cycles
Fri: Hopf bifurcation
Hilbert's problem on limit cycles
5. Week Lorenz: ODEs in higher dimensions
Mon: Differential equations in 3D
Lorenz, Forced oscillators etc.
Wed: The attractor in the Lorenz system
Lyapunov functions
Fri: Strange attractors
The notion of a fractal.
6. Week Birkhoff: billiards
Mon: Billiards and the variational setup
Existence of periodic points
Wed: An example of a chaotic billiard
Fri: Polygonal billiards
7. Week Hedlund: cellular automata
Mon: Curtis-Hedlund-Lyndon theorem
Wed: Topological entropy
Special solutions
Fri Attractors
Higher dimensional automata
8. Week Mandelbrot: maps in the complex plane
Mon: Mandelbrot and Julia sets
Some topological notions
Wed: Connectivity of Mandelbrot set
Fri: Iterations of quaternions
Complex Henon map
9. Week Bernoulli: subshifts of finite type
Mon: Bernoulli shift
Wed: Subshifts of finite type
Fri: Entropy
Normal numbers and randomness
10. Week Weyl: dynamical systems in number theory
Mon: Unique and strict ergodicity
Wed: Continued fractions
Fri: Diophantine problems
11. Week Poincare: many body problems
Mon: The equations of the n-body problem
Wed: Integrals and the solution of the 2 body problem
The Sitnikov problem
Fri: The role of singularities
12. Week Einstein: geodesic flows
Mon: Geodesic flows on the torus
Wed: Integrability and examples
Fri: Wave fronts and Huygens principle
Caustics
13. Week Review.
Mon: Review
Wed: Open problems in dynamical systems
Fri: Overview of projects
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