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 and records of streamed lectures are collected in a separate course PM.


The schedule of the course is in TimeEdit.


Lectures and exercises



Topics, notions, theorems, methods

Links to lecture notes, to recommended exercises, references





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.



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 with vector fields and phase portraits



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)

Download Exercises on linear autonomous ODE


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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

Shorter list with exercises at the Lecture 4 and home exercises



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.

Lecture notes: Introduction and autonomous linear systems
Exercises on linear autonomous ODEs
with exercises on exponents of matrices and on Jordan' matrices  with some solutions.



Boundedness and limit properties of the norm of the matrix exponent: LaTeX: \left|\right|exp\left(At\right)\left|\right|
Corollary 2.13, p. 36

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

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



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


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

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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.

Non-homogeneous linear systems of ODEs.
Variation of constant formula (Duhamel formula) for non-homogeneous linear equation,  the case with 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 than one in the book)
Stability of stationary points by linearization. Simple criteria.
Corollary 5.29, p.195,


Lecture notes on stability by linearization.




Formulation of the Grobman-Hartman theorem.

Exercises on stability by linearization.
Homogeneous linear non-autonomous ODEs.
Transition matrix function
Lemma 2.1, p.24; Corollary 2.3, p. 26

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

Exercises 5.20, 5.21, 5.22, 5.23.

§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



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

§2.1.2 Solution space

Lecture notes linear systems of ODE with variable coefficients and Floquet theory

 Exercises on non-autonomous linear systems.


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Two lectures



 Abel - Liouville's formula.  Proposition 2.7 part (2) . 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).
Example on calculation of monodromy matrix.        

Reflections on main ideas of Floquet theory.

§2.3, Floquet theory, §2.4

Lecture notes linear systems of ODE with variable coefficients and Floquet theory
Lecture notes linear systems of ODE with variable coefficients and Floquet theory.



Theorem on the structure of a transition matrix for
linear systems with periodic coefficients.
Th. 2.30, p. 53 Logarithm of a matrix. Prop. 2.29, p.53

Existence of the logarithm of a matrix
Spectral mapping theorem. Th. 2.19, mainly for f(x)=exp(x), and f(x)=log(x)
Floquet multipliers p.48.
Floquet's theorem on zero limit and on boundedness of solutions
to linear systems with periodic coefficients. Th. 2.31, p. 54.

Lecture notes linear systems of ODE with variable coefficients and Floquet theory

§2.3, Floquet theory, examples

Alternative proof to the existence of matrix logarithm

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

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Examples and exercises on periodic linear systems.
Existence of periodic solutions. Prop. 2.20, p.45


§2.3, Floquet theory, examples

Download exercises on linear periodic systems.
Exercise 2.9, p. 33
Exercise 2.16, 2.17, p. 47.



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.

Extension of bounded solutions. Lemma 4.9, p. 110; Cor. 4.10, p. 111.

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,p. 15;  1.4,1.5, p. 18-19
Exercise 4.2, 4.3, p. 109;
Exercise 4.4, p. 110
Exercise 4.8*,p. 114-115  




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 openness 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.

§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

The second modeling project - home assignment is available at Canvas

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Positive, negative semi-orbits.
Positively invariant sets. p. 141,
Omega limit points, omega - limit sets, 4.6.2, p. 141,

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.

User guide on  invariant and limit sets.

Download problems on invariant sets with answers

Examples on transition maps and limit sets: Exercise 4.16, p. 140, Example 4.37, p. 142, see lecture notes for solutions.

§4.7.1  Poincare- Bendixson theorem,
Exercise 4.21, p.158



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's 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)

§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


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
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

§4.7.2, First integrals and periodic orbits p. 161

§5.1 Lyapunov stability theory

Exercise 5.16, p. 188, Exercise 5.17, ,189
Download  problems on stability by
Liapunovs method with answers

Lecture notes with proofs to Lyapunov's stability and instability theorems


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Two lectures



Elementary introduction to LaSalle's invariance principle

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.

Asymptotic stability by "weak" Lyapunov's functon. Th. 5.15, p. 183.
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, 5.10.


Examples and exercises on stability and instability
by Lyapunov functions.
Exercises on application of
LaSalle's invariance principle

Matlab code for drawing a phase portrait in plane.

Download problems on Lyapunovs functions
from old exams

Deadline for the second project-home assignment is the 22-d of May

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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.

  Updated lecture notes on Banach's contraction principle and the Picard Lindelöf theorem.



Examples of bifurcations.

Exercises: Picard iterations.
Exercises on contraction principle.

Repetition of key ideas and methods in the course.
Preparation to examination

Exercises with solutions and hints
for Banach's contraction principle


Repetition of key ideas and methods in the course.
Preparation to examination

 Lecture notes on main techniques studied in the course with examples from exams

Examples of theoretical questions to the exam.


Examination. Monday, 1-st of June, 8:30

Home examintion as an assignment in Canvas

during Zoom meeting for control.



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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 2-3 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 2-d 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:

  1. Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as e-book from Chalmers library.
  2. 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.


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Course summary:

Date Details Due