SoSe 14: Differentialgleichungen I - Dynamical Systems
Bernold Fiedler
Information for students
Please find more information here: BMS-Course Dynamical Systems.
Additional information / Pre-requisites
This lecture is intended for students of mathematics or physics, including teachers, from semester 3.
Direct access to thesis projects: bachelor, master, dissertation.
There will be direct access to thesis projects: bachelor, master, dissertation.
The language of the recitation sessions will be English.
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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 first two semesters:
- Preliminaries: some calculus in Banach space
- Flows - differential equations - iterations
- Lyapunov functions and limit sets
- Planar flows and Nietzsche's dwarf
- Flows on tori and the "devil's staircase"
- Stable and unstable manifolds
- Shift dynamics: coding "chaos"
- Hyperbolic structure and the "butterfly effect"
- Ergodicity
- Shadowing
- Center manifolds
- Singular perturbations
- Normal form theory
- Averaging and "invisible chaos"
- The beauty of symmetry breaking
- A zoo of local bifurcations
- Genericity
- Takens embedding: dynamics without a model
- Global bifurcations and topological invariants
- Scientific Understanding of pictures
Depending on preferences of participants, the third semester may alternatively give an introduction to infinite-dimensional dynamical system, including certain partial and delay differential equations.
closeSuggested reading
- 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.
14 Class schedule
Regular appointments