Nonlinear Dynamics at the Free University Berlin

Winter 2020/2021

BMS-Course Dynamical Systems

Prof. Dr. Bernold Fiedler

Recitation sessions: Alejandro Lopez


Schedule, Wintersemester 2020/2021

Tuesdays and Thursdays, 10:15-12:00
ONLINE COURSE!
Live classes will take place via Zoom.
If you are planning to attend, we would be really grateful if you let us know by joining the KVV site of the course, so that we can have a rough idea of how many students to expect.
Information on how to join the lectures will be released via KVV. Additional materials will be provided for download.

Pass Criteria

Solve correctly at least 25% of the assignments. Hand in solution attempts for at least 50% of the assignments. Present a correct solution to an assignment on the blackboard in the recitation session at least once. Pass the written exam.

Written Exam

The written exam will take place on Tuesday February 23, 10:00-12:00 Berlin time.
More information in the KVV site of the course!

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 Geschichte(n) der Dynamik.

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 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. Although the course is offered as part 2 in the series, having a very basic knowledge on ODE theory should suffice in order to follow it.

Here is an outline of the first semester:

Last Semester:

  1. Existence and uniqueness of solutions of ordinary differential equations
  2. Flows, differentiablility and first integrals
  3. Linear differential equations
  4. Omega-limit sets and Lyapunov functions
  5. Planar flows and the Poincaré-Bendixson theorem
  6. Forced oscillations
Semester 2:

  1. Autonomous and forced oscillations
  2. Torus flows
  3. Stable and unstable manifolds
  4. Shift dynamics
  5. Hyperbolic sets
  6. Center manifolds
  7. Normal forms
  8. Genericity and Takens embedding

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.

Tutorials

Wednesdays 16:00-18:00, via WebEx! More information in the KVV site.

Homework 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.

  1. Assignment, due 19.11.2020 (PDF)
  2. Assignment, due 26.11.2020 (PDF)
  3. Assignment, due 03.12.2020 (PDF)
  4. Assignment, due 10.12.2020 (PDF)
  5. Assignment, due 17.12.2020 (PDF)
  6. Christmas exercises, due 11.01.2021 (PDF)
  7. Assignment, due 21.01.2021 (PDF)
  8. Assignment, due 28.01.2021 (PDF)
  9. Assignment, due 04.02.2021 (PDF)
  10. Assignment, due 11.02.2021 (PDF)
  11. Assignment, due 18.02.2021 (PDF)

Dynamical Systems II: basic questions

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