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Welcome to the course MMG511 (for GU)  /  MVE160 (for Chalmers).
The schedule for the course can be found via  webTimeEdit.
Alexei Heintz (examiner), room L2034,
Course literature

-- The course is decribed in the course syllabus, which states the learning goals of the course. From a practical point of view, the course consists of lectures, exercise classes and compulsory modeling assignments. In the exercise classes, some problems will be solved at the blackboard by a teacher, but all students are expected to participate actively. Study of each chapter will be followed by a homework to be checked by the teacher. Solution of some specified extra complicated problems will give extra bonus points for the course.
--A more detailed plan for the course will be regulary updated and extended. A List of Definitions, Theorems, and Typical Problems for Examination with page numbers is avalaible HERE
-- I will try to make notes after lectures and comments on exercises available on the home page.
Preliminary programme

Topic, chapter
Comments, recommended exercises
  18/3 §1.2-1.3  Introduction to ODE. Ideas and methods with examples of 1-d equations. Extensibility of solutions.  Example with non-unique solutions.
   Most problems in Chapter I are relevant
  19/3 §1.3-1.6 Theorem 1.3. Uniquness of solutions. Logistic growth equation. Qualitative analysis. Autonomous case with and without harvesting.
1.1, 1.3-1,9, 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
Easter and examination weeks 
  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.
§3.1-3.2. Matrix exponential,

  11/4 §3.1-3.2. 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

 15/4 §3.1-3.2 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, non-autonomous 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 e-mail to pose questions on the home assignment. 
 22/4 Exercises on linear systems. 
Ch.2  The contraction mapping theorem.
 23/4 Ch.2. Picard-Lindelöf theorem; existence, uniqueness,
Lecture notes on fixed points in Banach spaces, existense and uniqueness of solutions to ODE are HERE
 25/4 Picard-Lindelöf theorem; Alternative proof.

 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, 2.6-2.9, 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

 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

 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

 20/5 §7.1 Example from ecology. Volterra Lotka predator-pray 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

Modeling assingments
The purpose of the modeling assignments is to
1) give a deeper understanding of the connection between theory and modeling applications.
2) give a training in the use of mathematical software for modeling.
The assignments are compulsory, and the hand-ins (written reports) will be graded, and they will contribute 30% to the final marks.
The lab instructions are written in a pdf document.
  1. Material (in swedish) with short introduction to Matlab
  2. Holly More, MATLAB for Engineers
Course requirements
    Goals for the course are given in the course plan.
To pass this course you should pass the written exam and complete two modeling assignments, and participate acively in exercises as stated below.
Written reports on modeling assignments should be composed individually. However, it is allowed, and encouraged, to work together in pairs. In the reports, each student should then state with whom she/he has worked. The reports should be delivered to the examinor in electronic form, preferrably as a pdf-document, no later than the date of the written exam. The reports should be sufficiently complete to be read by somebody who does not have axcess to the lab instructions.
The final grade of the course is based on the marks of the written exam, which accounts for about 70% of the final grade, and the grade on the modeling assignments, which account for about 30% of the final grade. The participation in problem solving at the black board is required for passing the course, but that does not influence the final grade.

The written exam consists of problems to be solved and of theory questions. Some more inforamation about the written exam, and a list of important theory questions are given in the exam information list that will be supplied in the middle of the course.
During the exam the following aids are permitted: only pencil and eraser, no books, notes, calculators, no cell phones.
A List of Definitions Theorems and Typical Problems for Examination is avalaible HERE
(can be slightly updated before 20-th of May)
Tranings problems for examination can be downloaded HERE.

Examination procedures
At the following link you can find when exams are given: Schedule
During the exam only pencil and eraser are allowed.
Books, notes, calculators et.c. are not allowed.
At the exam, you should be able to show valid identification.
Before the exam, it is important that you report that you want to take the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, so sign up via GU's Study Portal.
Notice of result is obtained only by email via Ladok. (Not verbally at study expedition.) This is done automatically when the results are registered. Check that you have the right grades and score.

At the annual examination:
When it is practical a separate review is arranged. The date of the review will be announced here on the course website. Anyone who can not participate in the review may thereafter retrieve and review their exam on Mathematical sciences study expedition, Monday through Friday, from 9:00 to 13:00. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination:
Exams are reviewed and picked up at the Mathematical sciences study expedition, Monday through Friday, from 9:00 to 13:00. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.
Course evaluation
At the beginning of the course at leasttwo studenrepresentatives must be chosen to carry out the evaluation of the course together with the teacher. The course activities in GUL ( login in via Studentportalen) there is a form that can be used for evaluation. The evaluation is carried out as a discussion between the teacher and student representatives during the course and after the exam at a particular meeting with a rapport that filles in a standard form.