MVE162/MMG511 Ordinary differential equations and mathematical modelling
This page is under development and contains the program of the course: lectures and exercises, and modelling projects . Other information, such as learning outcomes, teachers, literature, lecture notes, records of streamed lectures, and examination are in a separate course PM.
Lecture notes and records of streamed lectures are collected in a separate course PM.
Program
The schedule of the course is in TimeEdit.
Lectures and exercises
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 
Må 
Exercises on solutions to linear autonomous ODE: generalized eigenspaces and general solutions. 
§2.1.3, §2.4 Shorter list with exercises at the Lecture 4 and home exercises 
On 
Jordan canonical form of matrix. Theorem A.9 , p. 268 and lecture notes. 
Lecture notes: Introduction and autonomous linear systems 

To 
Boundedness and limit properties of the norm of the matrix exponent: $$ 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

Lecture on scientific writing: Elin Götmark. 
Lecture notes (new!) after the lecture on scientific writing by Elin Götmark. You can also watch the lecture here (in Swedish). 


Fri. 04.03 13.00 
Exercises on phase portraits of autonomous linear systems in the plain and on exponents of matrices 
Download problems on autonomous linear ODEs and phase portraits Exercises on linear autonomous ODEs with updated calculations of exponents of matrices

The first home assignment  modeling project is published in Canvas. 

Easter and examination weeks 

W. 3 
Må 
In this lecture an introduction to the first project  home assignment is given. Nonhomogeneous linear systems of ODEs. 

On

Formulation of the GrobmanHartman theorem. Exercises on stability by linearization. 
Exercises on stability by linearizationNotes on stability by linearization for the pendulum with friction. §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 
Abel  Liouville's formula. Proposition 2.7 part (2) . Duhamel formula for nonautonomous linear systems. Th. 2.15, p.41 Reflections on main ideas of Floquet theory. 
§2.3, Floquet theory, §2.4 
Ons 
Theorem on the structure of a transition matrix for Existence of the logarithm of a matrix 
§2.3, Floquet theory, examples 

Deadline for the first assignment Modelling project 1 is on Monday the 4th of May. 

W. 5 
Må 
Examples and exercises on periodic linear systems. 
§2.3, Floquet theory, examples 
On. 
Nonlinear systems of ODEs, Chapter 4. Extension of bounded solutions. Lemma 4.9, p. 110; Cor. 4.10, p. 111. 
Lecture notes on existence and maximal solutions 


Tors. 
Limits of maximal solutions. Th. 4.11, p. 112. (escaping a compact property) 
§4.6.1, Flows and continuous dependence 
The second modeling project  home assignment is available at Canvas  
W. 6 
Må 
Positive, negative semiorbits. Examples on two 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 
Lecture notes on limit sets and Poincare Bendixson theorem. §4.7.1 Poincare Bendixson theorem, 
On 
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's 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) 
§4.7.3 Limit cycles. 

To 0514 
First integrals. Examples with Newton equation and pendulum. 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 
§4.7.2, First integrals and periodic orbits p. 161




W. 7 Two lectures 
Må 
Elementary introduction to LaSalle's invariance principle Main theorem on the properties of limit sets. LaSalle's invariance principle Th.5.12, p.180; Asymptotic stability by "weak" Lyapunov's functon. Th. 5.15, p. 183. 
$5.2 Invariance principles. 
Ons 
Examples and exercises on stability and instability 

Deadline for the second projecthome assignment is the 22d of May 

W. 8 
Må 
Banach spaces. C(I) Banach space. Fixed point problems. 
§A2. 


On 
Examples of bifurcations. Exercises: Picard iterations. 
Exercises with solutions and hints 

To 
Repetition of key ideas and methods in the course. 
Lecture notes on main techniques studied in the course with examples from exams 



Examination. Monday, 1st of June, 8:30 Home examintion as an assignment in Canvas during Zoom meeting for control.

Modelling projects
Two modeling projects  home assignments are obligatory and consist of a theoretical part that requires some mathematical reasoning and an implementation part including writing a simple Matlab code solving an ODE, graphical output, and analysis of numerical solutions and conclusions.
Each of you must write an own report but you are encouraged to work in small groups of 23 people discussing theoretical and programming problems.
The report must we written as a small scientific article that a person who did not study the course can understand. It must include: 1) theoretical argumentation, with necessary references 2) numerical results with graphical output and 3) interpretation of results, and names of all group members.
Second year batchelor students supply reports in Swedish and are advised to watch video (or streaming) of the the lecture on scientific writing that will be given by Elin Götmark on the 2d of April at 15:15. Students will get if necessary, feedback on scientific writing aspects of the preliminary version of the report to the first home assignment from Elin Götmark.
Students will supply reports and Matlab codes with clear comments via an assignment in Canvas to be checked by examiner Alexei Heintz, on mathematical qualities and (only the first project) by Elin Götmark, who will check the quality of scientific writing.
Grades for your reports on two home assignments will contribute 16% each to the final marks for the course.
One must pass independently both the exam and modeling projects to pass the course.
Reference literature:
 Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as ebook from Chalmers library.

Physical Modeling in MATLAB 3/E, Allen B. Downey
The book is free to download from the web. The book gives an introduction for those who have not programmed before. It covers basic MATLAB programming with a focus on modeling and simulation of physical systems.
Course summary:
Date  Details  Due 

