Summer 2018

BMS-Course Dynamical Systems Systems III:
Delay Differential Equations

Dr. Isabelle Schneider

Recitation sessions: Alejandro López Nieto

Schedule, Sommersemester 2019

Lecture:

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

Tutorials:

Alejandro López, Monday 16:00-18:00 A6/SR 009

Written exam / Klausur: the written exam will take place on Tuesday, July 9th, 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, July 12, in order to arrange a meeting.

Nachklausur: the resit exam will take place on Wednesday, October 16th, 10.15-11.45, A7/SR 140.

Pass Criteria

Solve correctly at least 40% 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

In diesem Semester beschäftigen wir uns mit der nichtlinearen Dynamik unter Einfluss von zeitlichen Verzögerungen. Zum einen wollen wir uns mit der Theorie solcher zeitverzögerten Differentialgleichungen beschäftigen, zum anderen aber auch einen Einblick in die vielen faszinierenden Phänomene bekommen, die durch den unendlich-dimensionalen Phasenraum entstehen. Zeitverzögerungen sind einerseits in vielen Systemen intrinsisch vorhanden, z.B. in optischen Systemen aufgrund der Signallaufzeiten, in biologischen Systemen mit Gedächtniseffekten und in komplexen Systemen in Wirtschaft, sozialen oder ökologischen Netzwerken, andererseits werden sie aber auch gezielt zur Kontrolle eingesetzt.

Vorausgesetzt werden gute Kenntnisse in gewöhnlichen Differentialgleichungen, wie sie zum Beispiel in der Vorlesung „Dynamical Systems 1“ im Sommersemester 2018 vermittelt wurden.

This semester we will study nonlinear dynamics under the influence of time delay. This course concerns itself with the the theory of delay differential equations and will offer a glimpse of the many fascinating phenomena that can precipitate in an infinite-dimensional phase space. Time delay appears as an intrinsic component of many systems, e.g., signal propagation in optical systems, biological systems with memory effects, complex economic models, and social or ecological networks. Time delay can also be artificially implemented into a system to control its behavior.

Prerequisites are some basic knowledge of ordinary differential equations (having attended a course on Dynamical Systems I, for example, should be more than enough).

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References

• J.K. Hale: Theory of Functional Differential Equations Springer, 1977.
• J.K. Hale, S.M. Verduyn Lunel: Introduction to Functional Differential Equations, Springer, 1993.
• H. Smith: An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2011.
• R.D. Driver: Ordinary and Delay Differential Equations, Springer, 1977.
• T. Erneux: Applied Delay Differential Equations, Springer, 2009.

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Homework assignments

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

1. Assignment, due April 17, 12:00 (PDF)
2. Assignment, due April 24, 12:00 (PDF)
3. Assignment, due May 2, 12:00 (PDF)
4. Assignment, due May 9, 12:00 (PDF) Notice that the mistake in part (iii) of problems 1 and 2 has been fixed!
5. Assignment, due May 16, 12:00 (PDF)
6. Assignment, due May 23, 12:00 (PDF)
7. Assignment, due May 31, 12:00 (PDF)
8. Assignment, due June 7, 12:00 (PDF)
9. Assignment, due June 28, 12:00 (PDF)

Basic questions

(PDF)