# MVE162/MMG511, Ordinary differential equations and mathematical modelling, 2018/19

## Latest news

Welcome to the course! The schedule for the course can be found in TimeEdit.

Solutions to the last exam on the 28-th 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 e-mail, but 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 which topics are difficult for you. I can then also modify my answers with further exposures if needed.

Students representatives for the course.
Chalmers:

MPENM   krieks@student.chalmers.se      Krister Ekström
TKTEM   carolina.stark@hotmail.com      Carolina Stark
TKTEM   lucasu@chalmers.se     Lucas Unnerfelt
guseveda@student.gu.se     David Evertsson

## 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.

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 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.

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

## Program

 Week Day Topics, notions, theorems, methods Links to lecture notes, to recommended exercises, references W.1 (13) Må 03-25 15:15 KA 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. The space of solutions to a linear ODE and it's dimension. 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 03-27 08:00 Euler Properties of matrix exponent. Lemma 2.10 (1),(3),(4),(5), p. 34; Examples of linear systems and their phase portraits. 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 an error in the Example 1.1 in the book is corrected, Matlab codes for illustrations (updated, works without pendulum function) To 03-28 08:00 Euler 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 Examples of solutions in the case when there is no basis of eigenvectors.$2.1.3 Autonomous systems (we consider it with more details and examples in lecture notes) W. 2 (14) M 04-01 15:15 KA Exercises on solutions to linear autonomous ODE: generalized eigenspaces and general solutions.   Real solutions to systems with real matrix having complex eigenvalues Th. 2.14, p. 38. Examples. §2.1.3, §2.4 Appendix 1, 2 On 04-03 08:00 Euler Jordan canonical form of matrix. Theorem A.9 , p. 268 and lecture notes. Exponent of Jordan matrix. Examples and exercises on Jordan matrices Exercises on calculations of exponents of matrices and fundamental matrix solutions for linear autonomous ODEs. Boundedness and limit properties of solutions to linear systems of ODEs. Corollary 2.13, p. 36 with exercises on Jordan' matrices and with some solutions To 04-04 08:00 Euler Stability and asymptotic stability of equilibrium (stationary) points. Definitions 5.1, p.169, 5.14, p.182. Phase portraits for linear autonomous ODEs in plane and their classification. Exercises on phase portraits of autonomous linear systems in plane To 04-04 15:15 HA2 Lecture on scientific writing: Elin Götmark. W. 3 (15) Må 04-08 15:15 KA In this lecture an introduction to the first project - home assignment is given. Stability of the equilibrium point in the origin for linear systems with constant coefficients. Propositions 5.23, 5.24, 5.25, p.189, p.190. We do it in a simpler way on the lectures. Theorem on existence and uniqueness of solutions to general I.V.P. Formulation of the Grobman-Hartman theorem. Exercises on stability by linearization. Exercises on stability by linearization Exercises 5.20, 5.21, 5.22 On 04-10 8:00 Euler Non-homogeneous linear systems of ODEs. 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. 5.27, p.193. Proof by Grönwall inequality in lecture notes. (simpler then one in the book) Stability of stationary points by linearization. Simple criteria. Corollary 5.29, p.195, - almost the same as Th.5.27. 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 04-11 8:00 Euler 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) Uniqueness of solutions to general systems of linear ODEs. Th. 2.5, p.28 Space of solutions to non-autonomous systems of linear ODEs 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 W. 4 (16) Two lectures Må 04-15 15:15 KA General variation of constant formula (Duhamel formula) for non-autonomous linear systems. 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) Tis 04-16 8:00 Euler Reflections on main ideas of Floquet theory. Theorem on the structure of a transition matrix for linear systems with periodic coefficients. Th. 2.30, p. 53Logarithm 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 multipliers p.48. Example on calculation of monodromy matrix in scalar case. §2.3, Floquet theory, examples Easter and examination weeks On Thirsday 25/4 at 11.00-12.00 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 (18) two lectures To 05-02 8:00 Euler Exercises 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 §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. Fred. 05-03 8:00 Euler Nonlinear systems of ODEs, 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 will be given 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. Lecture notes on existence and maximal solutions §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 Deadline for the first project Friday, 3-rd of May Second project - home assignment §1.1.1 necessary to read for carrying out the second project. W. 6 (19) Må 05-06 15:15 KA 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. On 05-08 8:00 Euler Positive, negative semi-orbits. Positively invariant sets. p. 141, Omega limit points, omega - limit sets, 4.6.2, p. 141, Examples on methods to find positively invariant sets. Periodic solutions of autonomous systems. §4.7.1, 4.7.2. 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 Examples on transition maps and limit sets: Exercise 4.16, p. 140, Example 4.37, p. 142, see lecture notes for solution. §4.7.1  Poincare- Bendixson theorem, Exercise 4.21, p.158 To 05-09 Euler Existence of an equilibrium point in a compact positively invariant set. Theorem 4.45, p. 150., 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. 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) 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 non-existence of periodic orbits §4.7.2, First integrals and periodic orbits p. 161 W. 7 (20) Må 05-13 15:15 KA Stability and asymptotic behavior of equilibrium points. Stability by Lyapunov functions. Th.5.2, p.170 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 §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 On 05-15 8:00 Euler General theory for omega - limit sets. 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 Exercise 5.7, 5.8 To 05-16 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 LaSalle's invariance principle Fr 18/5 W. 8 (21) Må 05-20 15:15 KA 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. Tis 22/5 Deadline for the second project-home assignment On 05-22 8:00 Euler Exercises: Picard iterations. Exercises on contraction principle. Repetition of key ideas and methods in the course. Preparation to examination To 05-23 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.

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.

## 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.

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

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

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,