# MVE162/MMG511, Ordinary differential equations and mathematical modelling, 2017/18

## Latest news

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

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 existence and maximal solutions

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

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   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

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 can be updated and complemented with references to particular examples and problems.

Most of exercises are available through links on the homepage.
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)

 Day Topics, notions, Links to lecture theorems, methods notes, to recommended exercises, references Week 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 (we consider it with more details in lecture notes) Download Exercises on linear autonomous ODE 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. updated 17 April,   The example on the Grobman - Hartman theorem  application is updated with analysis of the second equilibrium point. Download Exercises on stability by linearization 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 Periodic solutions of autonomous systems. §4.7.1, 4.7.2. 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) 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. 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 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. Invariance principles. 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 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.

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.