Nonlinear Dynamics at the Free University Berlin

Winter 2010/2011

BMS-Course Dynamical Systems

Prof. Dr. Bernold Fiedler

Recitation session: Dr. Stefan Liebscher

Schedule, Winter 2010/2011

Tuesday & Thursday, 10.15-12.00, seminar room 032, Arnimallee 6 (Pi building)
Recitation session:
Monday 14.15-16.00 (Juliette Hell), seminar room 053, Takustraße 9
Tuesday 8.30-10.00 (Stefan Liebscher), seminar room 032, Arnimallee 6
Written examination:
Tuesday, February 15, 10.15-11.45, seminar room 032, Arnimallee 6 (Pi building)
Written examination (resit):
Wednesday, April 20, 16-18, seminar room 032, Arnimallee 6 (Pi 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 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 course:

Last semester

  • Preliminaries: some calculus in Banach space
  • Flows - differential equations - iterations
  • Lyapunov functions and limit sets: the pessimism of decreasing energy
  • Planar flows and Nietzsche's dwarf

This and next semesters

  • Flows on tori and the "devil's staircase"
  • Stable and unstable manifolds: what is a continental divide?
  • Shift dynamics: coding "chaos"
  • Hyperbolic structure and the "butterfly effect"
  • Ergodicity: a static look at dynamics
  • Shadowing: errors which don't matter
  • Center manifolds: when hyperbolicity fails
  • Singular perturbations: do differential-algebraic models make any sense?
  • Normal form theory: let there be symmetry
  • Averaging and "invisible chaos"
  • The beauty of symmetry breaking
  • A zoo of local bifurcations
  • Genericity: to hell with mathematical degeneracy
  • Takens embedding: dynamics without a model
  • Global bifurcations and topological invariants
  • 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
Saratov Group of Theoretical Nonlinear Dynamics: Plykin Attraktor
Yves Coudene: Pictures of Hyperbolic Dynamical Systems (Notices of the AMS)

Homework assignments, Winter 2010/2011

  • voluntary problems given at the end of last semester (PDF)
  • 1st assignment, due Oct 28, 2010 (PDF)
    corrected version of the first problem, deadline extended to Nov 04, 2010.
  • 2nd assignment, due Nov 04, 2010 (PDF)
  • 3rd assignment, due Nov 11, 2010 (PDF)
  • 4th assignment, due Nov 18, 2010 (PDF)
  • 5th assignment, due Nov 25, 2010 (PDF)
  • 6th assignment, due Dec 02, 2010 (PDF)
  • 7th assignment, due Dec 09, 2010 (PDF)
  • 8th assignment, due Dec 16, 2010 (PDF)
  • voluntary problems, due Jan 07, 2011 (PDF)
  • 9th assignment, due Jan 13, 2011 (PDF)
  • 10th assignment, due Jan 20, 2011 (PDF)
  • 11th assignment, due Jan 27, 2011 (PDF)
  • 12th assignment, due Feb 3, 2011 (PDF)
  • 13th assignment, due Feb 10, 2011 (PDF)

Please use the appropriate boxes, Arnimallee 3, upstairs. Solve at least 2 problems (per assignment) in groups of two. Minimal requirement: average of 50% (of the 2 problems per assigment).

Basic questions

Archive Summer 2010

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