This page includes information about aim and learning outcomes, teachers, literature, examination, examination procedures, old exams and course evaluation. The program for all teaching sessions can be found on a separate page (Course syllabus).

Aim and learning outcomes

The aim and learning outcomes of the course can be found in the course plan.

Changes compared to the last occasion

In spring 2020 the course will be given as a distance course. We have already several tools to carry out it in this way. They are:

i) complete lecture notes for the course divided in chapters, that will be available from Canvas as pdf files from the very beginning, pdf files with exercises, answers and some solutions, old exams with complete solutions.

ii) Piazza platform for communicating questions, discussions. Piazza lets write very complicated formulas in latex even on the phone. Another advantage with Piazza is that questions and answers can be seen by the whole class. Questions can be posed anonymously. This channel will be available most of the time during the day. All participants registered in Canvas have got an invitation to Piazza.

iii) Probably some particular home exercises-assignments will be suggested to keep students on the trek of the course.

There is a plan to stream and (or) record lectures using existing lecture notes as a ground. Following alternatives are on the list:

iv) To stream and record lectures via internet using standalone Zoom platform according the the schedule. Here participants could pose questions as at a normal lecture. Recorded lectures will be available also later via links to YouTube put on this page in Canvas after lecture notes.

Additional Zoom meetings are possible to arrange if necessary.

It is a technically demanding task and a situation requiring much concentration both for students and for the teacher.

Teachers

Examiner: Alexei Heintz (Geynts), email: heintz(at)chalmers.se

Teaching assistants: Elin Götmark, elin(at)chalmers.se

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Course literature

Logemann, Hartmut, Ryan, Eugene P
Ordinary Differential Equations.
Analysis, Qualitative Theory and Control
Springer-Verlag London 2014
ISBN 978-1-4471-6397-8
The course book is avalable as electronic book at Chalmers' library.

Alternative book: One can try to use another book presenting some material on some other parts of the topic as Hamiltonian systems and discrete dynamical systems.
Gerald Teschl: Ordinary Differential Equations and Dynamical Systems, which can be purchased at The American Mathematical Society . A version of the book is available for free download from the author's web page.

A collection of lecture notes and exercises for the course (will be updated during the course).

Introduction and autonomous linear systems
Exercises on linear autonomous ODE, general solutions, Jordan's matrix, and matrix exponents.

Updated method for calculations of exponents of matrices

Shorter list with exercises (Links to an external site.) at the Lecture 4 and home exercises
Material on classification of phase portraits in plane.
Additional problems on autonomous linear ODE and phase portraits.

Lecture notes on stability by linearization.
Notes on stability by linearization for the pendulum with friction.
Exercises on stability by linearization.

Lecture notes on general linear systems of ODEs with variable coefficients and Floquet theory.
Material with proof and formulas for the matrix logarithm
Exercises on non-autonomous linear systems.
Exercises on linear periodic systems.

Lecture notes on existence and maximal solutions

Lecture notes on limit sets and Poincare - Bendixson theorem
Short user guide on invariant and limit sets
Lecture notes on Bendixson's criterion on non-existence of periodic orbits
Problems on invariant sets with answers
Exercises on periodic solutions and limit cycles from old exams

Lecture notes on Lyapunov's stability and instability theorems
Lecture notes on omega-limit sets and LaSalle's invariance principle with applications

Exercises on stability by Liapunov's method with answers
Problems on Lyapunov's functions from old exams

Exercises on application of LaSalle's invariance principle

Lecture notes on Banach's contraction principle and the Picard Lindelöf theorem.

Lecture on main techniques studied in the course with examples from exams

Examples of theoretical questions to the exam.

A link about Scottish cafe in Lvow, Poland, where Stefan Banach met with his colleagues -
famous mathematicians such as Schauder, Steinhaus, Saks, Borsuk, Ulam, and discussed solved, unsolved, and even probably unsolvable problems, that were in the way of discussions written in a huge notebook eventually became known as the Scottish Book.

Links to recorded lectures in YouTube will be put here

Lecture 1. Introduction. The dimension of the solution space to an autonomous linear system.LinkLecture 2 . Generalized eigenspaces and eigenvectorshttps://youtu.be/OYdbENLneGQ Lecture 3. The structure of general solutions to linear autonomous ODEshttps://youtu.be/tdSBVyfja9o Lecture 4. Exercises on solutions to linear autonomous ODEshttps://youtu.be/g9bncU4A7NI Lecture 5. Jordan matrices. Calculation of exponents of matrices.https://youtu.be/Yuo7hRlEGXo Lecture 6. Limit properties of the norm of exp(At)https://youtu.be/GMhhAp3ypYg Lecture on scientific writing. You can watch the lecture here (in Swedish).

Additional exercises in phase portraits and exponents of matrices.https://youtu.be/cVHKUQLzV-k Lecture 7. Stability by linearization.https://youtu.be/6HzrAWrOxwA Lecture 8. Examples on stability by linearization.

Linear non-autonomous systems of ODEs. Transition matrix function.

https://youtu.be/PLnGzFR-AbM Lecture 9. Chapman - Kolmogorov relations. Fundamental matrix solution.

https://youtu.be/nD8Vri1Y9XE Lecture 10. Abel - Liouville's formula. Periodic linear systems . Monodromy matrix.https://youtu.be/5WFrbbvxdRQ Lecture 11. Floquet theory. Logarithm of a matrix.

https://youtu.be/HGfyTCPF3dA Lecture 12. Exercises on periodic linear systems of ODEs.https://youtu.be/knwY5DK7hCU Lecture 13. Existence theorems and properties of maximal solutions.https://youtu.be/PjapASVDSaQ Lecture 14. Maximal solutions escaping compacts. Transfer mapping.https://youtu.be/iycW-NIfXSk Lecture 15. Positively invariant sets. Omega - limit sets. Poincare - Bendixson theoremhttps://youtu.be/vC1M_IHrkd4 Lecture 16. Exercises on Poincare Bendixson theoremhttps://youtu.be/81Wj5L8IbvQ Lecture 17. Bendixson's criterium. Lyapunov functions. First four.https://youtu.be/zoz7isW-Iv4 Lecture 17. Second hour. Asymptotic stability. Regions of attraction.https://youtu.be/jLlWB2vy6UM Lecture 18. Main theorem on omega limit sets. LaSalle's invariance principle with applications to asymptotic stability.https://youtu.be/M6M5Qi63zm0 Lecture 19. Examples and exercises on Lyapunov's functions and LaSalle's invariance principle.https://youtu.be/lk6ZFIQOYNw Lecture 20. Banach's contraction principle. Picard-Lindelöf theorem.https://youtu.be/-1ZJkKF_DBA Lecture 21. (first hour) Exercises on Picard-Lindelöf theorem https://youtu.be/6aKwb5Trrhc Lecture 21. (second hour)Exercises on Banach's contraction principle.

Introduction to bifurcations and Hopf bifurcation.https://youtu.be/cQpThR6YW48 Lecture 22. Overview of the course and examples with techniques and ideas.https://youtu.be/sP6JGh_4dZE

English-Swedish mathematical dictionary.

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Examination

The final grade for the course is based on the marks for the written exam, which account for 68% of the final grade, and on marks for two modeling assignments, which account for 16% each of the final grade. For those who have not passed the exam, points for the projects/assignment will be kept up to the moment when you will pass the exam.

The dates, times and places for the exams are in the student portal.

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Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply to exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at the University of Gothenburg.

Before the exam, it is important that you sign up for the examination. You sign up through Ladok.

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok.

At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination:
Exams are reviewed and retrieved at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

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Course evaluation

At the beginning of the course, at least two student representatives should have been appointed to carry out the course evaluation together with the teachers. The evaluation takes place through conversations between teachers and student representatives during the course and at a meeting after the end of the course when the survey result is discussed and a report is written.

Guidelines for Course evaluation in Chalmers student portal.

Student representatives

The following randomly chosen students have been appointed.

It is not evident that they really attend the course. We might need to appoint some more.

GU
TKTEM Rasmus.Durge@gmail.com Rasmus Durgé
TKIEK
TKTEM

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Old exams

A detailed list of Definitions, Methods, Theorems, and Typical Problems with proofs required at the exam marked.

Examples of theoretical questions to the exam.

Solutions to the exam in august 2013 can be downloaded HERE.
Solutions to the exam in june 2013 can be downloaded HERE.
Solutions to the exam on the august 25, 2014 are HERE
Suggestion for solutions to the exam on the June 2, 2014 is HERE
Solutions to the exam on the april 1, 2015 UPDATED on 2016.05.26 are HERE
Solutions to the exam in june 2015 are HERE
Solutions to the exam in august 2015 are HERE
Suggestions to solutions for problems in the last exam 2016.05.30 are HERE
Suggestions to solutions for problems in the last exam 2016.08.22 are HERE
Suggestions to solutions for problems in the exam 2017.01.03 are HERE
Suggestions to solutions for problems in the exam 2017.05.29 are HERE
Suggestions to solutions for problems in the exam 2017.08.21 are HERE
Solutions to the last reexamination on August 29, 2018, can be dowloaded HERE
Solutions to the exam on May 28, 2018, can be downloaded HERE.
Here are solutions to the exam on January 3-rd, 2018.
Solutions to the reexamination on January 7, 2019, can be downloaded HERE
Solutions to the exam on the 3-rd of June 2019 can be downloaded HERE

Solutions to the exam on the 1-st of June 2020 are HERE.

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Program

The program can be found on the Course syllabus.

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