Nonlinear Dynamics at the Free University Berlin

Summer 2011

BMS-Course Dynamical Systems

Prof. Dr. Bernold Fiedler

Recitation session: Dr. Stefan Liebscher

Schedule, Summer 2011

Tuesday 10.15-14.00, seminar room 032, Arnimallee 6 (Pi building)
Recitation session:
Wednesday 14.00-16.00, seminar room 140, Arnimallee 7 (rear building)
Written examination:
Tuesday, July 12, 10.15-11.45, seminar room 032, Arnimallee 6 (Pi building)
Written examination (resit):
Tuesday, August 30, 10.15-11.45, seminar room 130, Arnimallee 3 (rear 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 semesters

  • 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
  • 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?

This semester

  • 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.

Homework assignments, Summer 2011

  • 1st assignment, due Apr 22, 2011 (PDF)
  • 2nd assignment, due Apr 29, 2011 (PDF)
  • 3rd assignment, due May 06, 2011 (PDF)
  • 4th assignment, due May 13, 2011 (PDF)
  • 5th assignment, due May 20, 2011 (PDF)
  • 6th assignment, due May 27, 2011 (PDF)
  • 7th assignment, due June 03, 2011 (PDF)
  • 8th assignment, due June 10, 2011 (PDF)
  • 9th assignment, due June 17, 2011 (PDF)
  • 10th assignment, due June 24, 2011 (PDF)
  • 11th assignment, due July 01, 2011 (PDF)
  • 12th assignment, due July 08, 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, Winter 2010/2011

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