MVE162 / MMG511 Ordinary differential equations and mathematical modelling Spring 21
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 from 2020 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. Matrix exponential and general solution to a linear autonomous system. 
Appendix A.1, 
On 
Properties of matrix exponent. Lemma 2.10 (1),(3),(4),(5), p. 34; 
Lecture notes to first 6 lectures. 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 



Easter and examination weeks 29 March  9 April 

W. 2 
Må 
Exercises on solutions to linear autonomous ODE: generalized eigenspaces and general solutions. 
§2.1.3, §2.4 Shorter list with exercises for 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. 
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. 
Record of the lecture on scientific writing (in Swedish)  
The first home assignment  modeling project to be published in Canvas. 

W. 3 
Må 
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 
On

In this lecture an introduction to the first project  home assignment is given. Nonhomogeneous linear systems of ODEs. 
Introduction to modelling project. Lecture notes on stability by linearization.


To 
Stability of stationary points by linearization. Simple criteria. Formulation of the GrobmanHartman theorem. Exercises on stability by linearization.

Exercises on stability by linearizationNotes on stability by linearization for the pendulum with friction.




W. 4

Må 
Homogeneous linear nonautonomous ODEs. 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) 
§2.1.1 Homogeneous linear systems §2.1.2 Solution space 
Ons 
Abel  Liouville's formula. Proposition 2.7 part (2) . Duhamel formula for nonautonomous 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). Example on calculation of monodromy matrix. Reflections on main ideas of Floquet theory. 
§2.3, Floquet theory, §2.4



Tors 0429 8:00 
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 5th 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 The bonus assignment on applications of Floquet theory is available on Canvas 

W. 6 Two lectures 
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 homeomorphic to a ball. 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. 



W. 7 
Må 
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 §5.1 Lyapunov stability theory Download problems on stability by

Ons 
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. Theorem 5.22 , p. 188, on global asymptotic stability. 
$5.2 Invariance principles. Lecture notes on omegalimit sets and LaSalle's invariance principle with applications Exercise 5.7, 5.8, 5.10. 


Tors. 0520 8:00 
Examples and exercises on stability and instability


Deadline for the second projecthome assignment is the 27th 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. A detailed list of Definitions, Methods, Theorems, and Typical Problems (Links to an external site.) with proofs required at the exam marked. 
Lecture notes on main techniques studied in the course with examples from exams Examples of theoretical questions to the exam.




Examination. 0531. 8:30. Home examination 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 15th of April at 15:15. Students will get a feedback on scientific writing aspects of the preliminary version of the report to the first home assignment from Elin Götmark. Students will need to correct reports to the first home assignment according to this feedback.
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 
