Nonlinear Dynamics at the Free University Berlin

Summer 2014

BMS-Course Dynamical Systems

Prof. Dr. Bernold Fiedler

Recitation sessions: Bernhard Brehm, Jia-Yuan Dai, Anna Karnauhova

Schedule, Summer 2014

Tuesday, 10:15-14:00, Hörsaal B, Arnimallee 22
Bernhard Brehm, Monday 10:00-12:00 Seminarraum 032, Arnimallee 6
Jia-Yuan Dai, Wednesday 10:00-12:00 Seminarraum 009, Arnimallee 6
Anna Karnauhova, Friday 8:00-10:00 Seminarraum 025/026, Arnimallee 6
Written exam / Klausur:
July 15, 2014, Tuesday, 10:00-12:00, Hörsaal B, Arnimallee 22
Results (PDF), upload July 16, 2014, last update July 29, 2014
The exam can be viewed on Tuesday 12-14, August 5, 2014, Arnimallee 7, Room 132 (Bernhard Brehm's office)
Written exam (resit) / Nachklausur:
October 15, 2014, Wednesday, 10:00-12:00, Seminarraum E.31, Arnimallee 7
Results (PDF)
You may review your exams until 03.11.2014. If you want to do so, please send an email to Hannes Stuke.

Pass Criteria

Solve correctly at least 25% of the assignments. Hand in solution attempts for at least 50% of the assignments. Present a correct solution to an assignment on the blackboard in the recitation session at least once. Pass the written exam.


Students of mathematics or physics, including teachers, from semester 3. Direct access to thesis projects: bachelor, master, dissertation.


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 three 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 first two semesters:

Semester 1:

  1. Flows and differential equations
  2. First integrals, separation of variables, and the pendulum
  3. Examples and applications
  4. Existence, uniqueness, and differentiability
  5. Linear autonomous systems
  6. Omega-limit sets and Lyapunov functions
  7. Planar flows

Semester 2:

  1. Autonomous and forced oscillations
  2. Torus flows
  3. Stable and unstable manifolds
  4. Shift dynamics
  5. Hyperbolic sets
  6. Center manifolds
  7. Normal forms
  8. Genericity and Takens embedding
Depending on preferences of participants, the third semester may address current topics in finite-dimensional dynamics, or give an introduction to infinite-dimensional dynamical system, including certain partial and delay differential equations.


  • 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.
  • E. Zeidler: Nonlinear Functional Analysis and its Applications, Volume 1: Fixed-Point Theorems, Springer, 1998.

Homework assignments

Please form teams of two and hand in your joint solutions. Please note your name, Matrikelnummer and exercise session (either Anna, Bernhard or Jia-Yuan) on your solutions. Please, please, please staple your solutions together if you hand in multiple pages.

You can put your solutions into our boxes (Tutorenfächer) in the Arnimallee 3, first floor (just above the library).
  1. Assignment, due April 24 (PDF)
  2. Assignment, due May 2 (PDF)
  3. Assignment, due May 8 (PDF)
  4. Assignment, due May 15 (PDF)
  5. Assignment, due May 22 (PDF). Supplement: expliciteuler.m, lotka_volterra_example.m.
  6. Assignment, due May 30 (PDF)
  7. Assignment, due June 5 (PDF)
  8. Assignment, due June 12 (PDF)
  9. Assignment, due June 19 (PDF)
  10. Assignment, due June 26 (PDF)
  11. Assignment, due July 3 (PDF)
  12. Assignment, due July 10 (PDF)

Dynamical Systems I: basic questions

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