Summer 2010
BMSCourse Dynamical Systems
Prof.
Dr. Bernold Fiedler
Recitation session:
Dr.
Stefan Liebscher
Schedule, Summer 2010
 Lecture:
 Tuesday, 10.1514.00, lecture hall B (0.1.01), Arnimallee 14 (Physics building)
 Recitation session:
 Wednesday, 14.1516.00 (Stefan Liebscher), seminar room 053, Takustraße 9 (Computerscience building)
 Friday, 12.1514.00 (Eyal Ron), seminar room 032, Arnimallee 6
 Written examination:
 Tuesday, July 13, 1012, lecture hall B (0.1.01), Arnimallee 14 (Physics building)
 Results
 Written examination (resit):
 Wednesday, October 27, 1618, lecture hall B (0.1.01), Arnimallee 14 (Physics building)
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:
 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 differentialalgebraic 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
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,
McGillHill, 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 HoeneWronski
Homework assignments, Summer 2010
 assignment, due Apr 23, 2010, (PDF)
 assignment, due Apr 30, 2010, (PDF)
 assignment, due May 07, 2010, (PDF)
 assignment, due May 14, 2010, (PDF)
 assignment, due May 21, 2010, (PDF)
 assignment, due May 28, 2010, (PDF)
 assignment, due Jun 04, 2010, (PDF)
 assignment, due Jun 11, 2010, (PDF)
 assignment, due Jun 18, 2010, (PDF)
 assignment, due Jun 25, 2010, (PDF)
 assignment, due Jul 02, 2010, (PDF)
 assignment, due Jul 09, 2010, (PDF)
Please use the appropriate boxes, Arnimallee 3, upstairs.
voluntary problems (PDF)
Basic questions
Complete list of questions, the last 3 are not part of the written test.
