Day

Topics, notions, theorems, methods

Links to lecture notes, to recommended exercises, references

Må
03-20

15:15

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)
A simple version of Grönwall inequality, Lemma 2.4, p. 27, and uniqueness of solutions.

Matrix exponential and general solution to a linear autonomous system.
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.

On
03-22

08:00

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 to first 6 lectures. 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

To
03-23

08:00

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

Må
03-27

15:15

Exercises on solutions to linear autonomous ODE: generalized eigenspaces and general solutions.
 Chains of generalized eigenvectors and their linear independence.

§2.1.3, §2.4
Appendix 1, 2

Shorter list with exercises for the Lecture 4 and home exercises

On
03-29

08:00

Examples on complicated cases with chains of generalised eigenvectors.

Real solutions to systems with real matrix having complex eigenvalues
Examples.

Jordan canonical form of matrix. Theorem A.9 , p. 268 and lecture notes.

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.

To
03-30
08:00

Jordan canonical form of matrix. Theorem A.9 , p. 268 and lecture notes.
Exponent of Jordan matrix.

Boundedness and limit properties of the norm of the matrix exponent: $$
Corollary 2.13, p. 36

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

Deadline is the 28th of April.

Examination and Easter vacations  1-11 April

On
04-12

8:00

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.

Poincare diagram for phase portraits of linear autonomous ODEs in plane.

Material on classification of phase portraits in plane.
Poincare diagram for phase portraits of linear systems in plane

Matlab codes for illustrations with vector fields and phase portraits to non-linear ODE's

Matlab codes giving analytical solutions and drawing phase portraits for autonomous linear ODEs

Download problems on autonomous linear ODEs and phase portraits

Exercises on linear autonomous ODEs

with updated calculations of exponents of matrices

Tors
04-13

8:00

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.
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.

Formulation of stability of stationary points by linearization. Simple criteria. Corollary 5.29, p.195

Formulation of the Grobman-Hartman theorem.

Introduction to modelling project and lecture notes on stability by linearization.

Matlab codes for illustrations with vector fields and phase portraits

 Python code example for illustration with vector fields and phase portraits

To
04-13

15:15

Lecture on scientific writing (in Swedish) by Elin Götmark

 Lecture notes

Mon
04-17

15:15

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

Formulation of the Grobman-Hartman theorem.

Exercises on stability by linearization.

Exercises on stability by linearization

Notes on stability by linearization for the pendulum with friction.
Exercises 5.20, 5.21, 5.22, 5.23.

On
04-19

8:00

Exercises on stability by linearization.

 Exercises on stability by linearization.

Tors.
04-20

8:00

 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 solutions
§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
§4.3
Exercise 4.8*,p. 114-115  

Mon.
04-24

15:15

 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 

On
04-26

08:00

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

To.
04-27

8:00

Existence of an equilibrium point in a compact positively invariant set homeomorphic to a ball.
Theorem 4.45, p. 150.,
Exercises on Poincare-Bendixsons theory. 

Download exercises on
periodic solutions and limit cycles  HERE

On
05-03

8:00

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)

First integrals. Examples with Newton equation and pendulum.

§4.7.3 Limit cycles.

Lecture notes on Bendixson's criterion for non-existence 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
05-04

8:00

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
Region of attraction.

§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

Må
05-08

15:15

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 omega-limit sets and LaSalle's invariance principle with applications

Exercise 5.7, 5.8, 5.10.

Ons
05-10
8:00

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

Theorem 5.22 , p. 188, on global asymptotic stability.
Exponential stability by Lyapunov functions.Th.5.35, p.200

$5.2 Invariance principles.

Lecture notes on omega-limit sets and LaSalle's invariance principle with applications

Tors.

05-11

8:00

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

Må
05-15

15:15

 Homogeneous linear non-autonomous ODEs.
Transition matrix function  Lemma 2.1, p.24; Corollary 2.3, p. 26
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

Linear systems with periodic coefficients. Floquet's theory.
Shifting invariance property of transition matrix for periodic systems: (formula 2.31 , p. 45):  Φ(t+p,T+p)=Φ(t,T)

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
Appendix 2,3
Exercises 2.1, 2.2, pp. 22-23
Exercise 2.9, p. 33.
Exercises 2.13, 2.14, p. 42-43

§2.1.2 Solution space

§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.

 Exercises on non-autonomous linear systems.

Ons
05-17

8:00

  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 project-home assignment is the 25-th of May

Må
05-22

15:15

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.

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.

Alternative proof to the existence of matrix logarithm

On
05-24
8:00

 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.

Exercises: Picard iterations.
Exercises on contraction principle.

§A2.
  Updated lecture notes on Banach's contraction principle and the Picard Lindelöf theorem.

Exercises with solutions and hints
for Banach's contraction principle

Tors

05-25

8:00

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

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.