Nonlinear Dynamics at the Free University Berlin

Summer 2018

# BMS-Course Dynamical Systems

Prof. Dr. Bernold Fiedler

Recitation sessions: Hannes Stuke, Alejandro Lopez

## Schedule, Wintersemester 2018/2019

Lecture:
Tuesday, 10-12:00, Thursday, 10:00-12:00, A3/SR 130
Tutorials:
Alejandro López, Monday 14:00-16:00 A7/SR 140
Hannes Stuke, Wednesday 14:00-16:00 A3/SR 210
Written exam / Klausur: the written exam will take place in the last lecture week on Tuesday, February 12th, 10.15-11.45, A3/SR 130.
You can look up the exam (Klausureinsicht) on Thursday, February 14th, 10.15-11.45, A3/SR 130.
Grade: 5.0 4.0 3.7 3.3 3.0 2.7 2.3 2.0 1.7 1.3 1.0
Points 0-12 13-15 16-19 20-23 24-26 27-30 31-34 35-37 38-41 42-44 45-64
The results are here: (PDF)

## 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.

## 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.

Here is an outline of the first 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.

## Homework assignments

Please form teams of two and hand in your joint solutions.

1. Assignment, due 29.10.2018 (PDF)
2. Assignment, due 02.11.2018 (PDF)
3. Assignment, due 09.11.2018 (PDF)
4. Assignment, due 16.11.2018 (PDF)
5. Assignment, due 23.11.2018 (PDF)
6. Assignment, due 30.11.2018 (PDF)
7. Assignment, due 07.12.2018 (PDF)
8. Assignment, due 14.12.2018 (PDF)
9. Assignment, due 21.12.2018 (PDF)
10. Weihnachtsblatt, due 11.01.2019 (PDF)
11. Assignment, due 18.01.2019 (PDF)
12. Assignment, due 25.01.2019 (PDF)
13. Assignment, due 01.02.2019 (PDF) The typo in equation (3) has been fixed (the coefficients b1 and b2 were missing).

## Dynamical Systems II: basic questions

Last change: Feb. 13, 2019
 This page strictly conforms to the HTML4.01 standard and uses style sheets.