Nonlinear Dynamics at the Free University Berlin

Winter 2014/15

BMS-Course Dynamical Systems

Prof. Dr. Bernold Fiedler

!!! You have the possibility to check the corrections of your exam on Thursday 12.02.15 from 10-12 in room SR140. If you can not attend this appointment, you can sent an email within the next to weeks (until 24.02.2015) to Hannes Stuke in order to arrange an alternative date !!!

Schedule, Winter 2014/15

Tuesday, 10:15-14:00, Seminarraum 031, Arnimallee 6
Monday 10:00-12:00 Seminarraum 130, Arnimallee 3
Wednesday 10:00-12:00 Seminarraum E.31, Arnimallee 7
Written exam / Klausur:
February 10, 2015, Tuesday, 10:15-11:45
Results: (PDF)
Written exam (resit)/ Nachklausur:
April 13, 2015, Monday, 10:15-11:45 Seminarraum 140, Arnimallee7
Results: (PDF)
You have the possibility to discuss the results until the 29.04.2015. If you like to do that, please sent 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:

Last semester:

  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

This semester:

  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.)
  • J. Guckenheimer and P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 2003
  • 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.
  • A. Vanderbauwhede: Centre Manifolds, Normal Forms and Bifurcations in Dynamics Reported Volume 2, 1989.
  • 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 Hannes 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. Blatt, Abgabe am 23.10.2014, 14:00 (PDF)
  2. Blatt, Abgabe am 30.10.2014, 14:00 (PDF)
  3. Blatt, Abgabe am 06.11.2014, 14:00 (PDF)
  4. Blatt, Abgabe am 13.11.2014, 14:00 (PDF)
  5. Blatt, Abgabe am 20.11.2014, 14:00 (PDF)
  6. Blatt, Abgabe am 27.11.2014, 14:00 (PDF)
  7. Blatt, Abgabe am 04.12.2014, 14:00 (PDF)
  8. Blatt, Abgabe am 11.12.2014, 14:00 (PDF)
  9. Blatt, Abgabe am 18.12.2014, 14:00 (PDF)
  10. Weihnachtsblatt, Abgabe am 08.01.2015, 14:00 (PDF)
  11. Blatt, Abgabe am 15.01.2015, 14:00 (PDF) The first two exercises are due to the 22.01.2015. The second exercise was corrected.
  12. Blatt, Abgabe am 22.01.2015, 14:00 (PDF)
  13. Blatt, Abgabe am 29.01.2015, 14:00 (PDF)
  14. Blatt, Abgabe am 05.02.2015, 14:00 (PDF)

Dynamical Systems II: basic questions

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