MVE162 / MMG511 Ordinary differential equations and mathematical modelling
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 from 2022, solutions to old exams,
and records of streamed lectures from 2021 are collected in a separate course PM.
Program
The schedule of the course is in TimeEdit.
Lectures and exercises
Day 
Topics, notions, theorems, methods 
Links to lecture notes, to recommended exercises, references 
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 


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 
Examples on complicated cases with chains of generalised eigenvectors. Real solutions to systems with real matrix having complex eigenvalues Jordan canonical form of matrix. Theorem A.9 , p. 268 and lecture notes. 
Lecture notes:Introduction and autonomous linear systems 
To 
Jordan canonical form of matrix. Theorem A.9 , p. 268 and lecture notes. Boundedness and limit properties of the norm of the matrix exponent: $$ 

The first home assignment  modeling project is available in Canvas from the 3th of April. Deadline is the 28th of April. 

Examination and Easter vacations 111 April 

On 
Stability and asymptotic stability of equilibrium (stationary) points. Poincare diagram for phase portraits of linear autonomous ODEs in plane. 
Material on classification of phase portraits in plane. Matlab codes for illustrations with vector fields and phase portraits to nonlinear ODE's Matlab codes giving analytical solutions and drawing phase portraits for autonomous linear ODEs Download problems on autonomous linear ODEs and phase portraits 
Tors

Exercises on phase portraits of autonomous linear systems in the plain. In this lecture an introduction to to results necessary for the first project is given. Formulation of stability of stationary points by linearization. Simple criteria. Corollary 5.29, p.195 Formulation of the GrobmanHartman theorem. 
Introduction to modelling project and lecture notes on stability by linearization.

To 15:15 
Lecture on scientific writing (in Swedish) by Elin Götmark 



Mon 
Theorem on existence and uniqueness of solutions to general I.V.P. Nonhomogeneous linear systems of ODEs. Stability of stationary points by linearization. Simple criteria. Formulation of the GrobmanHartman theorem. Exercises on stability by linearization.

Exercises on stability by linearization Notes on stability by linearization for the pendulum with friction. 
On 
Exercises on stability by linearization.


Tors. 
Nonlinear systems of ODEs, Chapter 4. Existence and uniqueness theorems by Picard and Lindelöf. Prop. 4.15, p.115; on uniform Lipschitz property on the compact. Extension of bounded solutions. Lemma 4.9, p. 110; Cor. 4.10, p. 111. 
Lecture notes on existence and maximal solutions 



Mon. 
Limits of maximal solutions. Th. 4.11, p. 112. (escaping a compact property) 
§4.6.1, Flows and continuous dependence 
On 
Positive, negative semiorbits. Examples on two methods to find positively invariant sets. Periodic solutions of autonomous systems. §4.7.1, 4.7.2. 
Lecture notes on limit sets and Poincare Bendixson theorem. §4.7.1 Poincare Bendixson theorem, 
To. 
Existence of an equilibrium point in a compact positively invariant set homeomorphic to a ball. 




On 
Limit cycles. 4.7.3, p. 167. c Prop. 4.5.6, p. 165 on existence of limit cycles. First integrals. Examples with Newton equation and pendulum. 
§4.7.3 Limit cycles. Lecture notes on Bendixson's criterion for nonexistence of periodic orbits §4.7.2, First integrals and periodic orbits p. 161. First integral for pendulum without friction. First integral in a predator pray model. 
Tors 
Stability and asymptotic behavior of equilibrium points. Stability by Lyapunov functions. Th.5.2, p.170 Asymptotic stability by Lyapunov functions. Cor. 5.17, p.185 
§5.1 Lyapunov stability theory Download problems on stability by Lecture notes with proofs to Lyapunov's stability and instability theorems 


Må 
Proof of Cor. 5.17, p.185 on asymptotic stability by strong Lyapunov functions. Instability by Lyapunov functions. Th. 5.7, p. 174 Examples on using Lyapunov functions. Young inequality, Cauchy inequality. Elementary introduction to LaSalle's invariance principle 
Lecture notes on omegalimit sets and LaSalle's invariance principle with applications 
Ons 
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 
Tors. 0511 8:00 
Examples and exercises on stability and instability 

Må 
Homogeneous linear nonautonomous ODEs. Linear systems with periodic coefficients. Floquet's theory. Structure of the transition matrix for a time interval including a finite number of periods. (formula 2.32, p. 45 ): Φ(t+p,T)=Φ(t,0)[Φ(p,0) ]Φ(0,T) Monodromy matrix: Φ(p,0). 
§2.1.1 Homogeneous linear systems §2.3, Floquet theory, §2.4 Lecture notes linear systems of ODE with variable coefficients and Floquet theory Shorter version of lecture notes on linear systems with variable coefficients and Floquet theory. 
Ons 
Example on calculation of transition matrix. Motivating example and reflections on main ideas of Floquet theory. Floquet's theorem on factorisation of the transition matrix for a linear system with periodic coefficients. Th. 2.30, p. 53 
Lecture notes linear systems of ODE with variable coefficients and Floquet theory §2.3, Floquet theory, examples 
Deadline for the second projecthome assignment is the 25th of May


Må 
Logarithm of a matrix. Prop. 2.29, p.53 Existence of the logarithm of a matrix Examples and exercises on periodic linear systems. 
§2.3, Floquet theory, examples 
On 
Banach spaces. C(I) Banach space. Fixed point problems. Exercises: Picard iterations. 
§A2. Exercises with solutions and hints 
Tors 0525 8:00 
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.

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. 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 (or Python) 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.
It is not permitted to use chatGPT or similar tools (also of course not to plagiarize texts you find on the internet). The purpose of the exercise is for you to learn how to write a mathematical essay, and if you plagiarize, you won't learn it. In addition, there are often factual errors in texts generated by chatGPT. Suspected cases of cheating will be followed up and investigated.
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 
