
Day

Topic, chapter

Comments, recommended exercises

V1 
18/3 
§1.21.3 Introduction to ODE. Ideas and methods with
examples
of 1d equations. Extensibility of solutions. Example with
nonunique solutions.

Most problems in Chapter I are relevant

19/3 
§1.31.6 Theorem 1.3. Uniquness of solutions. Logistic growth
equation. Qualitative analysis. Autonomous case with and without
harvesting.

1.1, 1.31,9, 1.11, 1.14, 1.16, 1.27, 1.28, 1.32.
1.20, 1.30 (invesigate numerically: write a Matlab program and print
graphs for typical solutions)

21/3 
Logistic growth with periodic harvesting.
Poincare map. Demo exercises: 1.6, 1.7, 1.11, 1.15 
1.331.36. 
Easter
and
examination
weeks

V2 
08/4 
Suggested exercises from Chapter 1 are HERE
Lecture notes for Chapter 1
are HERE

Prepare for discussing the homework.


Facilitating
homework
on
Chapter
1
is
to
be
done
and
given
to
Alexei
before 16:00
Wednesday April 17: Solve Problems 1.16, 1.32, 1.33.
You can get a bonus point if
you solve a more complicated Problem 1.35.

In
particular:
Answer ALL questions and do analytical analysis of ALL
cases in Problem 1.16.
Give a formulation of the modified theorem and a
complete proof to it in Problem 1.32.

09/4 
Discussion of recommended exercises from Chapter 1.
§3.13.2. Matrix exponential, 

11/4 
§3.13.2.
invariant
subspaces,
Jordan
forms.
Functions
of
matrices.

Demo exercises: 3.1, 3.2, 3.11, 3.13, 3.14
Rec. exercises: 3.3, 3.4, 3.5, 3.8, 3.9, 3.10,
3.12


V3 
15/4 
§3.13.2 Linear autonomous systems; stable and
unstable
subspaces etc. Phase portraites in plane. Lecture
notes
on
linear
autonomous
systems are
HERE.
Summary
of
phase
protraits
in
plane in terms of Tr(A) and det(A)

Rec. exercises: 3.18,
3.24, 3.25, 3.26, 3.27, 3.30
Download problems on linear autonomous ODE
with answers HERE

16/4 
§3.4 Linear, nonautonomous systems. Principal matrix
solution. Theorem 3.9. Theorem 3.10, Lemma 3.15.
Lecture notes on general linear
system are HERE
§3.6. Linear systems with periodic coefficients.
Floquet's
theory.Monodromy matrix Theorem 3.15

Problems 3.39,
3.40, 3.41, 3.42,
3.43

18/4 
Stability of linear systems with periodic coefficients.
Logarithm of a matrix.
Lecture
notes
on
periodic
linear
systems and Floquet theory are HERE
Hill equation. Theorem 3.19. Matieu equation.
Introduction to the theory of inverted pendulum.

The first home
assignment:
Linear theory for inverted pendulum is HERE.

You
are
welcome
to
contact
the
teacher
in
his
office or by email to pose questions on the
home assignment. 
V4 
22/4 
Exercises on linear systems.
Ch.2 The contraction mapping theorem. 

23/4 
Ch.2. PicardLindelöf theorem; existence, uniqueness,
Lecture
notes
on
fixed
points
in
Banach
spaces,
existense
and uniqueness of solutions to ODE are HERE 

25/4 
PicardLindelöf theorem; Alternative proof.



V5 
29/4 
Gronwall inequality.
Stability with respect to data.
Theorem 2.8. Gronwall's nequality 2.38. p. 43
Extensibility of solutions. Lemma 2.14,
Corollary 2.15. Theorem 2.17.
Exercises on Chapter 2: 2.3, 2.5, 2.10, 2.14, 2.19
Lecture
notes
on
Gronwall's
inequality,
stablity with respect to data and
perturbations and extensibility are HERE

Rec. exercises: 2.1, 2.2, 2.4, 2.62.9, 2.12, 2.15,
2.18, 2.20, 2.22 



02/5 
Stability
of
solutions
by
linearization.
§3.7. Theorem 3.26, Corollary 3.27., p.101
Lecture notes on stability by linearization
are HERE

Download problems on stability by
linearization with answers


V6 
06/5 
§6.2, 6.3. Introduction to dynamical systems.
Notions of flow, invariant sets, limit sets.
§6.5 Notion of stability for fixed points.
Theorem 6.10 (the same as corollary 3.27 before)
Exercises on chapter 2 and on
stability by linearization.

Rec. exercises on dynamical systems: 6.1, 6.7, 6.8, 6.9,
Rec. exercises on stability of fixed points: 6.15, 6.16. 
07/5 
§6.5, 6.6. Stability by Liapunov functions.
Theorem 6.13 on stability. Examples.
Theorem on instability (absent in the book)

Start the second home
assignment:
Self excitable oscillations in
electrical circuits. Lienard equation. Poincare map.
An example of using EVENTS option in Matlab from Matlab help is HERE





V7 
13/5 
Krasovskii LaSalle Theorem 6.14 on asymptotic stability with
"weak" Ljapunovs function.
Proof of the Instability Theorem (absent in the book)
Exercises on Liapunov's functions.
Lecture notes on fixed points
stability are HERE.

Rec.
exercises
on
Liapunovs
functions:
6.18,
6.19
Download problems on stability by Liapunovs method with answers

14/5 
More various
exercises on stability of fixed points.
Periodic solutions of autonomous systems
(self excitable solutions). Poincare map for autonomous systems §6.4.
§7.2. Example from electrical egineering.
Lienard and Van der Pol equations.


16/5 
§7.3 Poincare Bendixson theorem.7.13
(without proof). Bendixson's criterium (Problem 7.11)
Exercises on periodic solutions.
Lecture notes on periodic solutions are HERE

Rec. exercises: 7.1, 7.9, 7.11, 7.12, 7.13
Download
problems
on
invariant
sets
with
answers


V8 
20/5 
§7.1 Example from ecology. Volterra Lotka
predatorpray
model and a model with limited growth..
Lecture
notes
on
examples
from
ecology are HERE

Rec.
exercises: 7.2, 7.5,
7.6 
21/5 
Preparation to examination 

23/5 
Preparation to examination
Lecture
notes
after
the
last
lecture with examples on principal matrix solution
and monodromy matrix are HERE

