19040 Lecture

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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Comments

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:

  1. Preliminaries: some calculus in Banach space
  2. Flows - differential equations - iterations
  3. Lyapunov functions and limit sets
  4. Planar flows and Nietzsche's dwarf
  5. Flows on tori and the "devil's staircase"
  6. Stable and unstable manifolds
  7. Shift dynamics: coding "chaos"
  8. Hyperbolic structure and the "butterfly effect"
  9. Ergodicity
  10. Shadowing
  11. Center manifolds
  12. Singular perturbations
  13. Normal form theory
  14. Averaging and "invisible chaos"
  15. The beauty of symmetry breaking
  16. A zoo of local bifurcations
  17. Genericity
  18. Takens embedding: dynamics without a model
  19. Global bifurcations and topological invariants
  20. 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.

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Suggested 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.
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14 Class schedule

Regular appointments

Tue, 2014-04-15 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Tue, 2014-04-22 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-04-29 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-05-06 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-05-13 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-05-20 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-05-27 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-06-03 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-06-10 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-06-17 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-06-24 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-07-01 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-07-08 10:00 - 14:00
Differentialgleichungen I - Dynamical Systems

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Tue, 2014-07-15 10:00 - 12:00
Differentialgleichungen I - Dynamical Systems (Klausur)

Comments:
Klausur

Lecturers:
Univ.-Prof. Dr. Bernold Fiedler

Location:
Hs B Hörsaal/Arnim 22 (Arnimallee 22)

Subjects A - Z