Nonlinear Dynamics at the Free University Berlin

Summer 2007, Winter 2007/2008

BMS-Course Dynamical Systems I, II

Prof. Dr. Bernold Fiedler

Recitation session: Dr. Stefan Liebscher


Schedule, Winter 2007/2008

Lecture:
Tuesday, 10.15-14.00, Seminar room 140, Arnimallee 7
Recitation session:
Monday, 14.15-16.00, Seminar room 140, Arnimallee 7

Topics

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

References

  • 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.


Homework assignments, Winter 2007/2008

  1. Due date: October 25, 2007 (PDF)
  2. Due date: November 01, 2007 (PDF)
  3. Due date: November 08, 2007 (PDF)
  4. Due date: November 15, 2007 (PDF)
  5. Due date: November 22, 2007 (PDF)
  6. Due date: November 29, 2007 (PDF)
  7. Due date: December 6, 2007 (PDF)
  8. Due date: December 13, 2007 (PDF)
  9. Due date: December 20, 2007 (PDF)
  10. Due date: January 17, 2008 (PDF)
  11. Due date: January 24, 2008 (PDF)
  12. Due date: January 31, 2008 (PDF)
  13. Due date: February 7, 2008 (PDF)
  14. Free extra problems (PDF)

Homework assignments, Summer 2007

  1. Due date: April 24, 2007 (PDF)
  2. Due date: May 8, 2007 (PDF)
  3. Due date: May 15, 2007 (PDF)
  4. Due date: May 22, 2007 (PDF)
  5. Due date: May 29, 2007 (PDF)
  6. Due date: June 5, 2007 (PDF)
  7. Due date: June 12, 2007 (PDF)
  8. Due date: June 19, 2007 (PDF)
  9. Due date: June 26, 2007 (PDF)
  10. Due date: July 3, 2007 (PDF)
  11. Due date: July 10, 2007 (PDF)
  12. Voluntary Problems (PDF)

Class picture, Winter 2007/2008

Class Pic 3

Class pictures, Summer 2007

Class Pic 1 Class Pic 2

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