D Y N A M I C A L    S Y S T E M S,    F a l l    2 0 1 1

Lectures: Monday, Wednesday and Friday, 10am-11am, in Science Center 411.

Lecturer: Paul Bourgade, office hours Wednesday and Friday 11am-12pm, you also can email me (bourgade@math.harvard.edu) to set up an appointment or just drop by (Science Center 341).

Course assistant: Tony Feng (tfeng@college.harvard.edu)

Course description: introduction to dynamical systems theory with a view toward applications. Topics include iterated maps, qualitative study of equilibria and attractors, bifurcation theory, ergodic theory. Applications include convergence speed of algorithms, dynamical systems from evolutionary biology, fractals.

Prerequisites: multivariable calculus. Some familiarity with measure theory is a plus.

Textbooks: Shlomo Sternberg's online lecture notes are the reference for most lectures.

Grading: problem sets (50%), midterm (15%) and a final project (35%).

A tentative schedule for this course is:

• Iterations, fixed points
• Aug. 31. Newton's method, the principle.
• Sep. 2. Newton's method, the convergence theorem.
• Sep. 7. The implicit function theorem, a first glimpse.
• Sep. 9. Basins of attraction, attractors, repellers.
• Sep. 12. The Schwarzian derivative.
• Sep. 14. Period 3 implies all periods.


• Bifurcations
• Sep. 16. The logistic map.
• Sep. 19. Local bifurcations I.
• Sep. 21. Local bifurcations II.
• Sep. 23. Feigenbaum's constant.
• Sep. 26. Lyapunov exponent.


• Conjugacy
• Sep. 28. Examples, monotone maps.
• Sep. 30. Chaos.
• Oct. 3. The quadratic transformation and the shift.
• Oct. 5. Diffeomorphisms of the circle I
• Oct. 7. Diffeomorphisms of the circle II
• Oct. 12. Midterm


• Contractions, fixed points
• Oct. 14. Complete metric spaces.
• Oct. 17. The contraction fixed point theorem.
• Oct. 19. The Lipschitz implicit function theorem.
• Oct. 21. Linearization near a hyperbolic point.
• Oct. 24. Invariant manifolds I.
• Oct. 26. Invariant manifolds II.
• Oct. 28. The Hausdorff metric, Hutchinson's theorem.
• Oct. 31. The classical Cantor set, the Sierpinski gasket.


• Ergodic theory
• Nov. 2. Definitions, examples.
• Nov. 4. Invariant measures.
• Nov. 7. Von Neumann.
• Nov. 9. Poincaré.
• Nov. 14. Birkhoff.
• Nov. 16. Liouville.


• Symbolic dynamics
• Nov. 18. Discrete dynamical systems.
• Nov. 21. Shifts of finite type.
• Nov. 28. Topological entropy.
• Nov. 30. The Perron Frobenius theorem.
• Dec. 2. Factors of finite shifts.

• Problem set 1.
• Problem set 2.
• Problem set 3.
• Problem set 4.

• December 7th 2011, 10am, The appearance of periodic cycles in measles epidemics, Nathan Georgette.
• December 7th 2011, 11am, Dynamics of the Brusselator, Monica Burgos.
• December 7th 2011, 12pm, Multi-moon orbiters and low thrust trajectories, Meghan Leddy.
• December 12th 2011, 10am, Chaos in the solar system, Peter Cha.
• December 12th 2011, 11am, Equilibrium points for a three body problem, Toan Phan.
• December 12th 2011, 12pm, KAM theory, Ian Humplik.