Winter 2019/2020

BMS-Course
Bifurcations: Theory and Applications

Prof. Dr. Bernold Fiedler

Recitation sessions: Alejandro López Nieto

Schedule, Wintersemester 2019/2020

Lecture:

Tuesday and Thursday 10-12:00, A3/SR 130

Tutorials:

Alejandro López, Thursday 8:30-10:00 (Sharp!) A7/SR 140

Written exam / Klausur: the written exam will take place on Thursday, February 13th, 10.15-11.45, A3/SR 130. The grades are already available (PDF).
If you wish to check your exam, please contact Alejandro before Friday, February 21, in order to arrange a meeting.
Nachklausur: the resit exam will be delayed until further notice.

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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.

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Course description

Bifurcations address responses of systems when parameters change. In experiments, that parameter may be a knob we turn up or down. Applications range from patterns in catalysis, embryogenesis, animal gaits, or fluid dynamics, to hysteresis, instabilities of high speed trains, catastrophic climate change, and symmetry breaking in cosmology. Mathematically, failure of the implicit function theorem is a key feature.

We will learn about topics and techniques like dimension reduction, elementary catastrophes, Euler's rod, group invariance and equivariance, Hopf bifurcation, subharmonic oscillations, time reversibility, and transitions to homoclinic chaos.

Prerequisite: some introductory course in differential equations, or equivalent courage. The course will adapt to your background, and fill in gaps when necessary. From semester 5.

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References

For references see the books of [Alligood, Sauer, Yorke], [V.I. Arnold], [Broer, Takens], [Chow, Hale], [Kielhöfer], [Yuri Kuznetsov], [Guckenheimer, Holmes], [Marsden, McCracken], [L.P. Shilnikov, A.L. Shilinikov, Turaev], [Vanderbauwhede], and [Wainwright]. But do not read all of that!

Slides from the talk at TU Berlin 21.10.19 (Slides)

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Outlook: seminars, bachelor theses, master's theses, and PhD thesis work using Dynamical Systems methods.

Homework assignments

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

1. Assignment, due October 29, 12:00 (PDF)
2. Assignment, due November 6, 12:00 (PDF)
3. Assignment, due November 13, 12:00 (PDF)
4. Assignment, due November 20, 12:00 (PDF)
5. Assignment, due November 27, 12:00 (PDF)
6. Assignment, due December 4, 12:00 (PDF)
7. Assignment, due December 11, 12:00 (PDF)
8. Assignment, due December 18, 12:00 (PDF)
9. Assignment, due January 15, 12:00 (PDF)
10. Assignment, due January 22, 12:00 (PDF)
11. Assignment, due January 29, 12:00 (PDF)
12. Assignment, due February 5, 12:00 (PDF)

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1. Extra Assignment, due November 13, 12:00 (PDF)
2. Extra Assignment, due December 4, 12:00 (PDF)
3. Weihnachtsblatt, due January 15, 12:00 (PDF)

Basic questions

(PDF)

Notes:

Norm continuity versus strong continuity