Latest news
Welcome to the course! The schedule for the course can be
found in TimeEdit.
Links to solutions of the old exams are added at the end of
the home page.
Solutions to the last
exam on May 28 updated
on May 29, can be downloaded
HERE.
A detailed list of
Definitions, Methods, Theorems, and Typical Problems
with proofs required at the exam marked.
A collection of lecture notes and exercises
for 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 non-uniqueness of solutions with phase
portraits to some non-linear 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 linear systems of ODE with variable coefficients
and Floquet theory.
Material
with proof and formulas for 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 updated on May 15
Lecture notes on
omega-limit sets and LaSalle's invariance principle with
applications updated on May 16
Exercises
on stability by
Liapunov's method with answers
Problems
on Lyapunov's functions
from
old exams
OLD 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.
Students are encouraged to
ask questions at lectures and by e-mail.
To create an environment for common discussions of
mathematical questions, we will use the course's common
web-based forum at Piazza.com. Our forum is called MVE162. All
participants in the course have been offered by e-mail the
opportunity to follow the forum.
An advantage
of the Piazza-based 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 what things
are difficult for you. I can then also modify my answers with further
exposures if needed.
Students representatives for the course.
Chalmers:
TKTEM
hogbda@student.chalmers.se
DAVID HÖGBERG
TKTEM
osjoh@student.chalmers.se
OSCAR JOHANSSON
MPENM
themis@student.chalmers.se
THEMIS MOULIAKOS
TKTEM
hannper@student.chalmers.se
HANNA PERSSON
TKTEM
voerik@student.chalmers.se
ERIK VON BRÖMSSEN
GU:
Teachers
Course coordinator: Alexei Heintz (Geynts), email: heintz(at)chalmers.se
Teaching assistant: Elin Götmark, elin(at)chalmers.se
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
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 can be updated and complemented with references to
particular examples and problems.
Lecture notes will be available for several parts of the course.
One can try to use another book presenting larger material on the topic.
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.
Program
Structure of the course
-
- Introduction to ODE. Ideas and methods. Formulation of basic existence and uniqueness results for the initial value problem.
- Linear ODE with constant coefficients and Jordan matrices. Stability of solutions.
- Phase portrait. Classification of phase portraits in plane.
- General theory for linear ODE with variable
coefficients.
- Linear ODE with periodic coefficients. Floquet theory. Stability of solutions.
- Several examples from ecology and physics are discussed through the course.
- Two projects - assignments on modeling some of the
examples from physics and ecology.
- The basic existence and uniqueness results for the
initial value problem with proofs. Dependence of solutions
on data.
- Basic notions for dynamical systems: flow, invariant
sets, limit sets, sets of attraction.
- Stability of fixed points and Liapunovs stability theory. Invariance principle by LaSalle.
- Periodic solutions to nonlinear ODE. Poincaré-Bendixon theorem (without proof).
Plan of lectures and exercises. Key notions and theorems are printed in bold characters. (will be updated)
| Week |
Day |
Topics, notions,
theorems, methods |
Links to lecture
notes, to recommended exercises, references |
|---|---|---|---|
| W.1 (12) |
Må 03-19 15:15 KE |
Course subject, structure, goals. Notion
of I.V.P. for ODE. Integral form of an ODE. Phase plane, examples of orbits, equilibrium points, periodic orbits, Linear ODE with constant coefficients (autonomous) Matrix exponential and general solution to a linear autonomous system. A simple version of Grönwall inequality, Lemma 2.4, p. 27, and uniqueness of solutions. Properties of matrix exponent. Lemma 2.10 (1),(3),(4),(5), p. 34; |
Appendix A.1, $2.1.3 Autonomous systems (we consider it with more details in lecture notes) Exercises 2.10, 2.11, p. 35, 2.12, p. 38. |
| On 03-21 08:00 KE |
Examples of linear systems and their
phase portraits. Method to find a basis for real solutions in the case with complex eigenvalues. Case with only one linearly independent eigenvector. Generalized eigenspaces and eigenvectors. Invariance of the generalized eigenspaces under the action of matrix A and exp(At) |
Lecture notes:
Introduction and linear autonomous systems updated 11 April, an error in the Example 1.1 is corrected, Matlab codes for illustrations (updated, works without pendulum function) |
|
| To 03-22 08:00 Euler |
Structure of the general solution to
linear ODE with constant coefficients;Th. 2.11; p.35
Examples of solutions in the case when there is no basis of eigenvectors. Exercises on solutions to
linear autonomous ODE: generalized
eigenspaces and general solutions
|
$2.1.3 Autonomous systems
Download Exercises on linear
autonomous ODE(we consider it with more details in lecture notes) Solutions are updated with a complete solution to the problem 864 added after discussions at Piazza |
|
| Easter and examination weeks |
|||
| First project-home assignment ( link is here) | |||
| W. 2 (15) |
M 09-04 15:15 KE |
Repetition. Jordan canonical form of matrix. Theorem A.9 , p. 268 and lecture notes Exponent of Jordan matrix. |
§2.1.3, §2.4 Appendix 1,2 |
| On 11-04 08:00 KE |
Examples and exercises on Jordan
matrices
Boundedness and limit properties of solutions to linear
systems of ODEs.Corollary 2.13, p. 36 Phase portraits for linear autonomous ODEs in plane and their classification. Exercises on phase portraits of
autonomous linear systems in plane
Real solutions to systems with real matrix having complex
eigenvalues Th. 2.14, p. 38. |
Lecture notes: Introduction and
autonomous linear systems updated 11 April with an argument explaining the connection between invariant subspaces and the block diagonal structure of a similar matrix. Download Exercises on linear autonomous ODE with exercises on Jordan' matrices with some solutions added 9 april Material on classification of phase portraits in plane. |
|
| To 12-04 08:00 Euler |
Exercises on calculation of exponents of matrices. (final lecture on linear autonomous systems) |
Exercises on linear autonomous
ODE are updated April 12, 13:00 with examples and detailed explanation of two methods of calculation of matrix exponent exp(A) Point out that the matrix in the exercise 6.4.64 was changed on the 13 of April |
|
| To 12-04 15:15 KE |
Lecture on scientific writing: Elin Götmark. | Lecture notes after the lecture on scientific writing by Elin Götmark. | |
| W. 3 (16) |
Må 16/4 15:15 KE |
This lecture is an
introduction to the first project -home assignment. Stability and asymptotic stability of equilibrium (stationary) points. Definitions 5.1, p.169, 5.14, p.182. Stability of the equilibrium point of the origin for linear systems with constant coefficients. Propositions 5.23, 5.24, 5.25, p.189, p.190. Examples. We did it simpler on the lectures Theorem on existence and uniqueness of solutions to general I.V.P. Formulation of Grobman-Hartman theorem. Exercises on stability by linearization. |
Lecture
notes on stability by linearization.
Download Exercises on stability by
linearizationupdated 17 April, The example on the Grobman - Hartman theorem application is updated with analysis of the second equilibrium point. Exercises 5.20, 5.21, 5.22 |
| On 18/4 8:00 KE |
Variation of constant
formula (Duhamel formula) for non-homogeneous
linear equation, in the case of constant
coefficients.Corollary 2.17, p. 43. Stability of equilibrium points for a linear autonomous system perturbed by a “small” nonlinear right hand side. Th. 5.7, p.193. Proof by Grönwall inequality in lecture notes. (simpler then in the book) Th. 5.31, p.196, - the same as Th.5.7. Stability of stationary points by linearization. Corollary 5.29, Homogeneous linear non-autonomous ODEs. Transition matrix function and fundamental matrix solution Lemma 2.1, p.24; Corollary 2.3, p. 26 |
§2.1.1 Homogeneous linear systems Appendix 2,3 Exercises 2.1, 2.2, pp. 22-23 Exercise 2.9, p. 33. Exercises 2.13, 2.14, p. 42-43 |
|
| To 19/4 8:00 Euler |
Example. Stability by linearization
for the pendulum with friction.
Grönwall's inequality. Lemma 2.4, p. 27 (we use and prove only a simple version with constant coefficient under the untegral) Uniqueness of solutions to systems of linear ODEs. Th. 2.5, p.28 Space of solutions and its dimension : Prop. 2.7 first statement , p.30. Example 2.2, p.26. Group properties of the transition matrix function (Chapman Kolmogorov relations): Corollary 2.6, p.29 Fundamental matrix solution for linear homogeneous ODE, Prop. 2.8, p. 33 |
§2.1.2 Solution space Notes on stability by linearization for the pendulum with friction. Exercises on non-autonomous linear systems. |
|
Second project-home assignment ( a link is here) |
|||
| W. 4 (17) |
Må 23/4 15:15 KE |
General variation of
constant formula (Duhamel formula) Th. 2.15, p.41 Linear systems with periodic coefficients. Floquet's theory. Property of transition matrix for periodic systems: formula (2.31) , p. 45 Φ(t+p,T+p)=Φ(t,T) Monodromy matrix: Φ(p,0) Theorem on the structure of a transition matrix for linear systems with periodic coefficients. Th. 2.30, p. 53 |
§2.3, Floquet theory,
§2.4 Lecture notes linear systems of ODE with variable coefficients and Floquet theory updated on april 29. Alternative proof to existence of matrix logarithm |
| On 25/4 8:00 KE |
Reflections on ideas of
Floquet theory. Logarithm of a matrix. Prop. 2.29, p.53 Spectral mapping theorem. Th. 2.19, mainly for f(x)=exp(x), and f(x)=log(x) Floquet multipiers p.48. |
||
| To 26/4 8:00 Euler |
Exercise on
periodic linear systems. Floquet's theorem on zero limit and on boundedness of solutions to linear systems with periodic coefficients. Th. 2.31, p. 54. Existence of periodic solutions. Prop. 2.20, p.45 Abel's formula. Formula 2.14 in Proposition 2.7, p.30 A sufficient criteria for the existence of an unbounded solution to a periodic system. Corollary 2.33, p.59. |
§2.3, Floquet theory,
examples Lecture notes linear systems of ODE with variable coefficients and Floquet theory updated on april 29. Download exercises on linear periodic systems. Exercise 2.9, p. 33 Exercise 2.16, 2.17, p. 47. |
|
| |
|||
| W. 5 (18) two lectures |
On 2/5 8:00 KE |
Example of a periodic
system: Kapitza pendulum and the Hill equation. Example 2.32, p. 55-56. Nonlinear systems of ODE, Chapter 4. Peano existence theorem Th. 4.2, p. 102 (without proof) Existence and uniqueness theorems by Picard and Lindelöf. Th. 4.17, p. 118 (for continuous f(t,x), locally Lipschitz in x), Th.4.22, p.122 (for piecewise continuous f(t,x), locally Lipschitz in x) (proof comes later, in the last week of the course). Prop. 4.15, p.115; on uniform Lipschitz property on the compact. Maximal solutions. Continuation of solutions. Existence of maximal solutions. Th. 4.8, p.108. |
§1.2.1, §1.2.3,
§4.1, Existence of solutions §4.2, Maximal asolutions §4.3, 4.4, Existence and uniqueness of solutions. Exercises 1.3,1.4,1.5, p. 18-19 Exercise 4.2, 4.3, p. 109; Exercise 4.4, p. 110 §4.3 Exercise 4.8*,p. 114-115 |
| To 3/5 8:00 Euler |
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. Openness of the domain and continuity of transition map. Theorem 4.29, Lemma 4.30, p. 129 (only idea of the proof is discussed) Autonomous differential equations §4.6 Flows, openness of domain and continuity. Example 4.33., p. 139. Theorem 4.34, p.139 (consequence of Th. 4.29, p. 129) |
§4.6.1, Flows and
continuous dependence §4.6.2, Limit sets $4.6.3, Equilibrium points and periodic points Exercise 4.15, p. 140, Exercise 4.16, p. 140, Exercise 4.17, p. 140 Lecture notes on non-linear systems. Existence, extension updated on May 5 (under construction) |
|
| To 3/5 |
Deadline for the first project-home assignment |
||
| W. 6 (19) two lectures |
Må 7/5 15:15 KE |
Transition property of
transition map.
Positive, negative semi-orbits. Positively invariant sets. p. 141, Existence of an equilibrium point in a compact positively invariant set. Theorem 4.45, p. 150., Limit points, limit sets, 4.6.2, p. 141, Poincare- Bendixson theorem 4.46, p. 151 (Only 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. updated 7 May (under construction) User guide on invariant and limit sets. Download problems on invariant sets with answers §4.7.1 Poincare- Bendixson theorem, Exercise 4.21, p.158 |
| On 9/5 8:00 KE |
Examples
on transition maps and
limit sets: Exercise
4.16, p. 140,
Example 4.37, p. 142,
Exercises on Poincare-Bendixsons 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. |
§4.7.3 Limit cycles. Download exercises on periodic solutions and limit cycles HERE |
|
| W. 7 (20) |
Må 14/5 15:00 KE |
Bendixson criterion for
non-existence of periodic solutions:
div(f) >0 or
div(f)<0 on a simply connected domain
in plane - without holes (after lecture notes)
Stability by Lyapunov functions. Th.5.2, p.170 First integrals and periodic orbits. §4.7.2. Prop. 4.54, p. 161 on periodic orbits by first integrals. Examples: Lotka Volterra eq., Example 4.55 Nonlinear pendulum: Exercise 4.23, p. 164. Stability and asymptotic behaviour
of equilibrium points.
Instability by Lyapunov functions. Th. 5.7, p. 174 Asymptotic stability by Lyapunov functions. Cor. 5.17, p.185 Region of attraction. Theorem 5.22 , p. 188, on global asymptotic stability. Exponential stability by Lyapunov functions.Th.5.35, p.200 |
Lecture notes on
Bendixson's criterion for non-existence of periodic
orbits §4.7.2, First integrals and periodic orbits p. 161 §5.1 Lyapunov stability theory Exercise 5.16, p. 188, Exercise 5.17, ,189 Lecture notes with proofs to Lyapunov's stability and instability theorems updated on May 14 |
| On 16/5 8:00 KE |
General theory
for omega - limit sets. Translation
property of autonomous systems. Theorem
4.26, pp. 126-127.
Invariance principles. Group property of the flow. Theorem 4.35, p. 140. (non-linear version of Chapman - Kolmogorov relations) Flows, openness of domain and Lipschitz continuity property. Theorem 4.34, p.139 (consequence of Th. 4.29, p. 129) Main theorem on the properties of limit sets. Omega- limit sets are connected and consist of orbits. Th. 4.38, p.143 LaSalle's invariance
principle Th.5.12, p.180;
we take the proof from the solution to Exercise 5.9, p. 312. Example 5.13, p. 181 |
$5.2 Invariance
principles. Lecture notes on omega-limit sets and LaSalle's invariance principle with applications updated on May 16 Exercise 5.7, 5.8 §5.6, 5.7 Linearization of nonlinear systems (repetition) |
|
| To 17/5 8:00 Euler |
Asymptotic
stability by "weak" Lyapunov's functon. Th.
5.15, p. 183. Examples and exercises on stability and instability by Lyapunov functions. Exercises on application of invariance principles |
Download
problems on stability by Liapunovs method with answers Download problems on Lyapunovs functions from old exams |
|
| Fr 18/5 |
|||
| W. 8 (21) |
Må 21/5 15:15 KE |
Banach spaces. C(I)
Banach space. Fixed point problems. Contraction mapping principle by Banach.Theorem A.25, p. 277 Lemma 4.21, p.121 Picard-Lindelöf existence and uniquness theorem with proof; Picard iterations Th. 4.22, p. 122. |
§A2. OLD lecture notes on Banach's contraction principle and the Picard Lindelöf theorem. |
| Tis 22/5 |
Deadline for the second project-home assignment | ||
| On 23/5 8:00 KE |
Exercises: Picard
iterations.
Exercises on contraction principle. |
Exercises with solutions
and hints for Banach's contraction principle |
|
| To 24/5 8:00 Euler |
Repetition of key ideas
and methods in the course. Preparation to examination |
||
Computer labs
Reference literature:
Learning MATLAB, Tobin A. Driscoll ISBN: 978-0-898716-83-2 (The book is published by SIAM).
Course requirements
The learning goals of the course can be found in the course
plan.
To pass this course you should pass the written exam and
complete two modeling projects/assignments.
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 pdf-document
and preferably before deadline.
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.
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) quality of
analytical work and understanding of the theory 3) quality of
numerical implementation, 4) quality of graphical illustrations.
Examination
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 will do this by 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 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.
Old exams
Here are solutions to the last exam on January 3-rd, 2018.
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
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