Nonlinear Dynamics at the Free University Berlin

Sommer 2016

# Differentialgleichungen I

PD Dr. Martin Väth

Exercises: Nicola Vassena

## Schedule, Summer 2016

Lecture:
Monday 12.15-14.00, HS 001, Arnimallee 3
Wednesday 12.15-14:00, HS 001, Arnimallee 3
Exercises:
Tuesday, 12.15-14.00, SR 130, Arnimallee 3 (Nikita Begun) NO TUTORIAL ON 19th APRIL
Friday, 10.15-12.00, 025/026, Arnimallee 6 (Nicola Vassena)

## News

Please register (in addition to the standard registration) into KVV. This is not a substitute for the standard registration and has no obligations; for instance, you need not be afraid to register into KVV if you are not sure yet whether you will join the course, but it will help us in organizing things.

Instructions to do this:

1. Go to the KVV homepage.
2. Login with Direkter KVV-Login with your ZEDAT account. (If you do not have a ZEDAT account and cannot easily get one, please ask after lecture or exercises.)
3. Go to Membership, choose Joinable Sites, choose the lecture (Basismodul: Differentialgleichungen I) and press the join button.

## Content

The lecture is an introduction to ordinary differential equations with an outlook on dynamical systems.

Many applications of mathematics in physics, biology, chemisty and other sciences, but also within mathematics itself, lead to ordinary differential equations or systems of differential equations. Very often, these equations cannot be solved „explicitly“, and even the numerical analysis can be very hard for long times or many parameters.

In an introduction in this topic the theoretical analysis of such equations is therefore especially important (besides some important cases in which an explicit analytic approach is possible). In the lecture a special emphasis is put on general methods for this theoretical analysis. Planned sections in the lecture are:

1. Elementary solution methods
2. Existence and uniquenes
3. Linear differential equations
4. Dependence of the solution on the data
5. Outlook on dynamical systems, e.g.:
Flowbox theorem, stability, Ljapunuv functions, stable and unstable manifolds, periodic solutions and their stability

## Inhalt

Die Vorlesung bietet eine Einführung in gewöhnliche Differentialgleichungen mit Ausblicken auf dynamische Systeme.

Viele Anwendungen der Mathematik in Physik, Biologie, Chemie und anderen Wissenschaften, aber auch innerhalb der Mathematik selbst, führen auf gewöhnliche Differentialgleichungen oder Differentialgleichungssysteme. Diese Gleichungen lassen sich oft nicht „explizit'“ lösen, und selbst die numerische Analyse kann für lange Zeiträume oder viele Parameter schwierig sein.

In einer Einführung in das Thema ist daher - neben wichtigen Fällen, in denen ein expliziter analytischer Zugang möglich ist - die theoretische Analyse solcher Gleichungen besonders wichtig. Die Vorlesung legt besonderen Wert auf allgemeine Methoden für diese Analyse. Geplante Kapitel sind u.a.:

1. Elementare Lösungsmethoden.
2. Existenz und Eindeutigkeit.
3. Lineare Differentialgleichungen.
4. Abhängigkeit der Lösungen von den Daten.
5. Ausblicke auf Dynamische Systeme, etwa:
Begradigungssatz, Stabilität, Lyapunov-Funktionen, Stabile und instabile Mannigfaltigkeiten, Periodische Lösungen und ihre Stabilität.

## Literature

1. W. Walter. Gewöhnliche Differentialgleichungen, Springer, 2000.
2. H. Amann. Ordinary differential equations. Walter de Gruyter, 1990.
3. G. Teschl. Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, 2012.
4. J. W. Prüss, M. Wilke. Gewöhnliche Differentialgleichungen und Dynamische Systeme, Birkäuser, 2010.
5. P. Hartman. Ordinary Differential Equations. Cambridge University Press, 2002.

## Dates for Exams (Klausuren)

The Klausur will be on July 13, 2016 during lecture.
Start of the Klausur already on 12:00

The Nachklausur will be on Monday, 17.10.2016, 14:00-16:00 (possibly longer) in room SR 032, Arnimallee 6

Dates to look up the Exam: October 19-November 2, 2016, Room 136, Arnimallee 7. If I should not be there, ask in the office 142 (or better fix a date with me e.g. by email).

You can bring one (A4) paper with you for the exam on which you can have written whatever you want. Please also bring empty paper and a pen with you! Nothing else is admissible (in particular: no calculators, computers, smartphones, books, lecture notes etc.)

## Exercises for summer 2016 and their finishing dates

For attending the final exam, every person needs to obtain 50% of the total points. Please form groups of not more than three people and hand in your joint solutions. Please note clearly your name, Matrikelnummer and your exercise session (either Nicola or Nikita) on your solutions. Please staple your solutions together if you hand in multiple pages. The last day in which you can hand in is Friday, you can put your solutions into Nicola Vassena's box (Tutorenfächer) in Arnimallee 3, ground floor (just beside the stairs). Please write the solutions to the exercises in English (because they are corrected by the Tutors)!
1. blatt01.pdf (German version) exercise01.pdf (English version), April 29, 2016
2. exercise02.pdf May 6, 2016
3. exercise03.pdf May 13, 2016
4. exercise04.pdf May 20, 2016
5. exercise05.pdf May 27, 2016
6. exercise06.pdf June 03, 2016
7. exercise07.pdf June 10, 2016
8. exercise08.pdf June 17, 2016
9. exercise09.pdf June 24, 2016
10. exercise10.pdf July 1, 2016
11. exercise11.pdf July 8, 2016

## Suggestions for some solutions of current exercises

The solutions are no longer available online.

## Manuscript

The manuscript is no longer available online.
Last change: Oct. 19, 2016
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