Nonlinear Dynamics at the Free University Berlin

Summer 2010

BMS-Course Dynamical Systems

Prof. Dr. Bernold Fiedler

Recitation session: Dr. Stefan Liebscher

Schedule, Summer 2010

Tuesday, 10.15-14.00, lecture hall B (0.1.01), Arnimallee 14 (Physics building)
Recitation session:
Wednesday, 14.15-16.00 (Stefan Liebscher), seminar room 053, Takustraße 9 (Computer-science building)
Friday, 12.15-14.00 (Eyal Ron), seminar room 032, Arnimallee 6
Written examination:
Tuesday, July 13, 10-12, lecture hall B (0.1.01), Arnimallee 14 (Physics building)
Written examination (resit):
Wednesday, October 27, 16-18, lecture hall B (0.1.01), Arnimallee 14 (Physics building)


Dynamical Systems are concerned with anything that moves. Through the centuries, mathematical approaches take us on a fascinating voyage from origins in celestial mechanics to contemporary struggles between chaos and determinism.

The two semester course, aimed at graduate students in the framework of the Berlin Mathematical School, will be mathematical in emphasis. Talented and advanced undergraduates, however, are also welcome to this demanding course, as are students from the applied fields, who plan to really progress to the heart of the matter.

Here is an outline of the course:

  1. Preliminaries: some calculus in Banach space
  2. Flows - differential equations - iterations
  3. Lyapunov functions and limit sets: the pessimism of decreasing energy
  4. Planar flows and Nietzsche's dwarf
  5. Flows on tori and the "devil's staircase"
  6. Stable and unstable manifolds: what is a continental divide?
  7. Shift dynamics: coding "chaos"
  8. Hyperbolic structure and the "butterfly effect"
  9. Ergodicity: a static look at dynamics
  10. Shadowing: errors which don't matter
  11. Center manifolds: when hyperbolicity fails
  12. Singular perturbations: do differential-algebraic models make any sense?
  13. Normal form theory: let there be symmetry
  14. Averaging and "invisible chaos"
  15. The beauty of symmetry breaking
  16. A zoo of local bifurcations
  17. Genericity: to hell with mathematical degeneracy
  18. Takens embedding: dynamics without a model
  19. Global bifurcations and topological invariants
  20. Scientific Understanding of pictures


  • K.T. Alligood, T.D. Sauer and J.A. Yorke: Chaos, Springer, 1997.
  • H. Amann: Ordinary Differential Equations, de Gruyter, 1990.
  • V.I. Arnold: Ordinary Differential Equations, Springer, 2001.
  • V.I. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, 1988.
  • W.E. Boyce and R.C. DiPrima: Elementary Differential Equations and Boundary Value Problems, Wiley, 5th edition, 1992.
  • S.-N. Chow and J.K. Hale: Methods of Bifurcation Theory, Springer, 1982.
  • E.A. Coddington and N. Levinson: Theory of ordinary differential equations, McGill-Hill, 1955.
  • P. Collet and J.-P. Eckmann: Concepts and Results in Chaotic Dynamics. A Short Course, Springer, 2006.
  • R. Devaney, M.W. Hirsch and S. Smale: Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2003.
    (This is the updated version of
    M.W. Hirsch and S. Smale: Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.)
  • Dynamical Systems I, D.K. Anosov and V.I. Arnold (eds.), Encyclopaedia of Mathematical Sciences Vol 1, Springer, 1988.
  • J. Hale: Ordinary Differential Equations, Wiley, 1969.
  • B. Hasselblatt, A. Katok: A First Course in Dynamics, Cambridge 2003.
  • P. Hartmann: Ordinary Differential Equations, Wiley, 1964.
  • A. Katok, B. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems, Cambridge 1997.
  • F. Verhulst: Nonlinear Differential Equations and Dynamical Systems, Springer, 1996.

F. Nietzsche: Also sprach Zarathustra III. Vom Gesicht und Rätsel.
Piotr Pragacz: Notes on the life and work of Jozef Maria Hoene-Wronski

Homework assignments, Summer 2010

  1. assignment, due Apr 23, 2010, (PDF)
  2. assignment, due Apr 30, 2010, (PDF)
  3. assignment, due May 07, 2010, (PDF)
  4. assignment, due May 14, 2010, (PDF)
  5. assignment, due May 21, 2010, (PDF)
  6. assignment, due May 28, 2010, (PDF)
  7. assignment, due Jun 04, 2010, (PDF)
  8. assignment, due Jun 11, 2010, (PDF)
  9. assignment, due Jun 18, 2010, (PDF)
  10. assignment, due Jun 25, 2010, (PDF)
  11. assignment, due Jul 02, 2010, (PDF)
  12. assignment, due Jul 09, 2010, (PDF)

Please use the appropriate boxes, Arnimallee 3, upstairs.

voluntary problems (PDF)

Basic questions

Complete list of questions, the last 3 are not part of the written test.

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