Summer 2007, Winter 2007/2008

BMS-Course Dynamical Systems I, II

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:

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?
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, 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.