Latest news
Welcome to the course! The schedule for the course can be
found in TimeEdit.
Solutions to the last exam on the 28th of
August 2019 can be downloaded HERE
Teachers
Course
coordinator: Alexei Heintz
(Geynts), email:
heintz(at)chalmers.se
Teaching
assistant: Elin
Götmark, elin(at)chalmers.se
PIAZZA FORUM
Participants in the
course are encouraged to pose questions by email, but to
create an environment for common discussions of
mathematical questions, we will use the course's common
webbased forum at Piazza.com. Our forum is called MVE162. All participants in the
course have been offered by email the opportunity to
follow the forum.
An advantage of the
Piazzabased forum is that it's easy to write
complicated formulas there, even on smartphone. Another advantage is that many students
often have similar questions and will benefit from
participating in discussions at the forum. Please ask your
questions there (it's good to be anonymous) and I will
answer as soon as possible. Students participating in the
forum can also try to answer questions. I will of course go through others answers
and edit if necessary.
If you ask a question and receive an answer from
me, it would be good if you could confirm in the
discussion if the answer was helpful or not. This makes it easier for me
to know which topics are difficult for you. I can then also
modify my answers with further exposures if needed.
Chalmers:
MPENM krieks@student.chalmers.se Krister Ekström
TKTEM carolina.stark@hotmail.com Carolina Stark
TKTEM lucasu@chalmers.se Lucas Unnerfelt
GU gussunly@student.gu.se Lydia Andersson
guseveda@student.gu.se David Evertsson
Course literature
Course
literature
Logemann,
Hartmut, Ryan, Eugene P
Ordinary Differential Equations.
Analysis, Qualitative Theory and Control
SpringerVerlag London 2014
ISBN 9781447163978
The course book is avalable as electronic book
at Chalmers' library.
A detailed
list of Definitions, Methods, Theorems, and Typical
Problems that are
studied in the course with references to page numbers in the
course book .
Proofs required at the exam are marked.
It is going to be the main check list during studies and for
preparation to the exam.
This list will be updated and complemented with references
to particular examples and problems.
Exercises are available
through links on the homepage.
Lecture notes for the course are available and include
many examples and exercises.
A collection of lecture notes and exercises for
the course from the year 2018 (will be updated during the
course).
Introduction
and autonomous linear systems
Exercises on
linear autonomous ODE, general solutions, Jordan's matrix,
and matrix exponents.
Material on
classification of phase portraits in plane.
Additional
problems on autonomous linear ODE and phase portraits.
Includes:
i) general introduction with examples of blow up of
solutions and nonuniqueness of solutions with phase portraits
to some nonlinear equations. ii) theory of linear sytems with
constant matrix (autonomous), exponent of a matrix, Grönwall's
inequality and uniquness proof, description of the space of
solutions, generalized eigenvectors, Jordan's canonical form
of the matrix. Exercises include examples on calculation of:
generalized eigenvectors, general solutions to I.V.P.,
Jordan's canonical form, and calculation of the exponent
of a matrix.
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
nonautonomous 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 nonexistence 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 omegalimit 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
NEW lecture notes on Banach's contraction principle
and the Picard Lindelöf theorem.
A link about Scottish
cafe in Lwow,
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.
Alternative book: One can try to use another book
presenting 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.
EnglishSwedish mathematical dictionary
Program
Week 
Day 
Topics, notions,
theorems, methods 
Links to lecture notes, to
recommended exercises, references 
W.1 
Må 
Course subject, structure, goals. Notion
of I.V.P. for ODE. 
Appendix A.1, 
On 
Properties of matrix exponent. Lemma
2.10 (1),(3),(4),(5), p. 34; 
Lecture
notes: Introduction and linear autonomous systems 

To 
Structure of the general solution to
linear ODE with constant coefficients;Th. 2.11; p.35 Examples of solutions to linear
autonomous ODE: generalized eigenspaces and
general solutions

$2.1.3 Autonomous systems 





W. 2 
M 
Exercises on solutions to linear
autonomous ODE: generalized eigenspaces and
general solutions. 
§2.1.3, §2.4 
On 
Jordan
canonical form of
matrix. Theorem
A.9 , p. 268 and lecture notes. Boundedness and limit properties of
solutions to linear systems of ODEs. 
Lecture notes: Introduction and
autonomous linear systems 

To 
Stability and asymptotic stability
of equilibrium (stationary) points. Exercises on phase portraits of autonomous linear systems in plane

Material
on classification of phase portraits in plane. Download
problems on autonomous linear ODEs and phase
portraits Matlab
codes giving analytical solutions and drawing
phase portraits for autonomous linear ODEs 

To HA2 
Lecture on scientific writing: Elin
Götmark. 
Lecture
notes (old) after the lecture on scientific
writing by Elin Götmark. 

First projecthome assignment  
W. 3 
Må 
In this lecture an introduction to the
first project  home assignment is given. 
Lecture
notes on stability by linearization. Exercises on
stability by linearization 
On Euler 
Nonhomogeneous
linear systems of ODEs. 
§2.1.1 Homogeneous linear systems 

To 
Grönwall's inequality. Lemma 2.4, p. 27
(we use and prove only a simple version of the
inequality with constant coefficient under the
integral) 
§2.1.2 Solution space 



W. 4 Two lectures 
Må 
General variation of constant formula
(Duhamel formula) for nonautonomous linear systems.
Th. 2.15, p.41 
§2.3, Floquet theory, §2.4 
Tis 
Reflections on main ideas of Floquet
theory. 
§2.3, Floquet theory, examples Alternative proof to the existence of matrix logarithm 

Easter
and examination weeks On Thirsday 25/4 at 11.0012.00 in MVF31 students can come and discuss language and structure of the first project with Elin Götmark. Mathematical questions can be posed to Alexei via PIAZZA or at the teaching time. 

W. 5 
To 
Exercises on periodic linear systems. 
§2.3, Floquet theory, examples 
Fred. 
Nonlinear systems of ODEs, Chapter 4.

Lecture
notes on existence and maximal solutions 




Second project  home
assignment §1.1.1 necessary to read for carrying out the second project. 

W. 6 
Må 
Extension of bounded solutions. Lemma 4.9,
p. 110; Cor. 4.10, p. 111. Limits of maximal solutions. Th. 4.11, p. 112. (escaping a compact property) On"global" extensibility of solutions for an ODE with a linear bound for the right hand side. Prop. 4.12, p.114, Examples 4.6, 4.7, p.108 on extensibility of solutions Transition map. Def. p.126. The openess of the domain and the continuity of transition map. Theorem 4.29, p. 129; Theorem 4.34, p.139 (autonomous case) Transition property of transition map. Th. 4.26, p.126; Th. 4.35, p.140 (autonomous case) Autonomous differential equations §4.6 Transition maps are called also flows in autonomous case. 
§4.6.1, Flows and continuous
dependence 
On 
Positive, negative
semiorbits. Examples on methods to find
positively invariant sets. Poincare Bendixson theorem 4.46, p. 151 (Only an idea of the proof is discussed). Applications of Poincare Bendixson theorem, p. 157 Example 4.57, p. 165

Lecture
notes on limit sets and Poincare Bendixson
theorem. §4.7.1 Poincare
Bendixson theorem, 

To 0509 Euler 
Existence of an
equilibrium point in a compact positively invariant
set. Theorem 4.45, p. 150., Exercises on PoincareBendixsons theory. Examples of periodic solutions from physics and ecology. Limit cycles. 4.7.3, p. 167. c Prop. 4.5.6, p. 165 on existence of limit cycles. Bendixson criterion for nonexistence of periodic solutions: div(f) >0 or div(f)<0 on a simply connected domain in plane  without holes (after lecture notes) First integrals and periodic orbits. Examples: Lotka Volterra eq., Example 4.55 Nonlinear pendulum: Exercise 4.23, p. 164. 
§4.7.3 Limit cycles. Download exercises on periodic solutions and limit cycles HERE Lecture notes on Bendixson's criterion for nonexistence of periodic orbits §4.7.2, First integrals and periodic orbits p. 161 

W. 7 
Må 
Stability and asymptotic behavior of
equilibrium points. Stability by Lyapunov functions. Th.5.2, p.170 

On 
General theory for omega
 limit sets. Main
theorem on the properties of limit sets. LaSalle's invariance principle
Th.5.12, p.180; 
$5.2 Invariance principles. 

To 
Asymptotic
stability by "weak" Lyapunov's functon. Th. 5.15, p. 183. 
Download
problems on stability by 

Fr 

W. 8 
Må 
Banach
spaces. C(I) Banach space. Fixed point problems. 
§A2. 
Tis 
Deadline for the second projecthome
assignment 

On 
Exercises: Picard iterations. 
Exercises with solutions and hints 

To 
Repetition of key ideas and methods in
the course. 

Computer labs
Reference literature:
Learning MATLAB, Tobin A. Driscoll ISBN: 9780898716832 (The book is published by SIAM).
Course requirements
The learning goals of the course can be found in the course
plan.
The learning goals of the course can be found in the course
plan.
Swedish second year students must write rapports on the projects in Swedish.
Exchange students and master students can write rapports in English.
Written reports on the modeling projects should be composed individually. However, it is allowed, and encouraged, to work together in pairs. In the reports, each student should then state with whom she/he has worked. Without this notification the report will not be accepted. The reports should be delivered to the examiner in electronic form, preferably as a pdfdocument and preferably before deadline.
Assignments
Instructions for projects  assignments will be available in
PING PONG and GUL. There will be put also grades for the
projects/asingments with short comments.
The reports should be written in a form of a small scientific
article sufficiently complete to be understood by somebody who
does not have access to the instructions. The questions posed in
each assignment must be clearly answered. The quality of the
reports to the modeling projects is estimated according to: 1)
the quality of the text and presentation, 2) the quality of
analytical work and theory understanding 3) the quality of
numerical implementation and graphical illustrations.
Examination
The final grade of the course is based on the marks of the written exam, which accounts for about 70% of the final grade, and the grades on two modeling assignments, which account for about 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.Examination procedures
In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.
Before the exam, it is important that you sign up for the
examination. If you study at Chalmers, you can do this from the
Chalmers Student Portal, and if you study at University of
Gothenburg, you sign up via GU's
Student Portal.
At the exam, you should be able to show valid identification.
After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.
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 reexamination:
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.
Old exams
A detailed list of Definitions, Methods, Theorems, and Typical Problems with proofs required at the exam marked.
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 3rd, 2018.
Solutions to the reexamination on January 7, 2019, can be downloaded HERE
Solutions to the exam on the 3rd of June 2019 can be downloaded HERE