NEWS:

 
ONLINE COURSE!
 
START: Tuesday, April 21 2020, 10:15-12:00.
Please see the details in the News section above and also check your emails (Zedat account).
Live classes will be attempted on webex zoom. If, and only if, webex fails miserably, all class materials will be provided for download in two parts per class, afterwards. Part 1 consists of a concise summary. The complementary part 2 will discuss each and every step of the summary by detailed screen and audio recording. It is strongly recommended to go through part 2 first, to take notes and digest the material at leisure. Do not procrastinate! Anytime later, and as a reminder, you may prefer and enjoy the brevity of part 1.
 
 

Pass Criteria

Solve correctly at least 25% of the assignments (see assignments section). Fachbereich and university have not reached a policy concerning final exams (Klausur), at yet. Information will be provided, in due time. In case such binding guidance does not become available, in due time, course credit will be based on active participation, only.

Audience

Students of mathematics or physics, including teachers, from semester 3. Direct access to thesis projects: bachelor, master, dissertation. Students interested in dynamical systems are also welcome to participate in the seminar (Hi)stories of Dynamics. Please note that topics will be distributed on April 23, 12 p.m., right after the second session of this course.
It will be possible to give your talk in English.

Topics

Dynamical Systems are concerned with anything that moves. The three semester course takes us on a fascinating mathematical voyage through the centuries: from origins in celestial mechanics to contemporary struggles between chaos and determinism. Examples include nonlinear oscillators, ecological models, the Lorenz equation, chemical and metabolic networks, and epidemic models.

Semester 1:

  1. Flows and differential equations
  2. First integrals and the nonlinear pendulum
  3. Examples, examples, examples
  4. Existence and uniqueness of solutions of ordinary differential equations
  5. Linear differential equations
  6. Omega-limit sets and Lyapunov functions
  7. Planar flows and the Poincaré-Bendixson theorem
  8. Forced oscillations

Assignments

Form teams of two, work on four problems per week, submit at least two, get one right each week (on average). Submission procedure: email a .pdf or .jpg file of your solutions to your assigned tutor, before the deadline. Be prepared to explain any of your solutions (no matter whether your own or the solution by your team partner!) during any tutorial, live, on webex.

A total of 40 points is needed to pass the course. Moreover, remember that you need to have presented at least once during the tutorial sessions.

Tutorials

Please choose one of the following:

Please make sure that you have a working microphone, and at best, the possibility to share the screen as well as a webcam.

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.
  • E. Zeidler: Nonlinear Functional Analysis and its Applications, Volume 1: Fixed-Point Theorems, Springer, 1998.