Ordinary Differential Equations and Dynamical Systems, 6 credits (TATA71)

Ordinära differentialekvationer och dynamiska system, 6 hp

Main field of study

Mathematics Applied Mathematics

Level

First cycle

Course type

Programme course

Examiner

Hans Lundmark

Director of studies or equivalent

Jesper Thorén

Available for exchange students

Yes
Course offered for Semester Period Timetable module Language Campus VOF
6KMAT Mathematics, Bachelor's Programme 3 (Autumn 2018) 2 3 Swedish/English Linköping v
6KMAT Mathematics, Bachelor's Programme (Computer Science) 3 (Autumn 2018) 2 3 Swedish/English Linköping v
6KMAT Mathematics, Bachelor's Programme (Modelling and Optimization in Economics) 3 (Autumn 2018) 2 3 Swedish/English Linköping v
6KMAT Mathematics, Bachelor's Programme (Mathematics) 3 (Autumn 2018) 2 3 Swedish/English Linköping v
6KMAT Mathematics, Bachelor's Programme (Applied Mathematics) 3 (Autumn 2018) 2 3 Swedish/English Linköping v
6CEMM Energy-Environment-Management M Sc in Engineering 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CMMM Mechanical Engineering, M Sc in Engineering 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics) 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics) 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics) 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics) 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics) 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYY Applied Physics and Electrical Engineering, M Sc in Engineering 7 (Autumn 2018) 2 3 Swedish/English Linköping v
6CYYY Applied Physics and Electrical Engineering, M Sc in Engineering (Applied Mathematics) 7 (Autumn 2018) 2 3 Swedish/English Linköping v

Main field of study

Mathematics, Applied Mathematics

Course level

First cycle

Advancement level

G2X

Course offered for

  • Mathematics, Bachelor's Programme
  • Energy-Environment-Management M Sc in Engineering
  • Mechanical Engineering, M Sc in Engineering
  • Applied Physics and Electrical Engineering - International, M Sc in Engineering
  • Applied Physics and Electrical Engineering, M Sc in Engineering

Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.

Prerequisites

Linear Algebra, Calculus one variable, Calculus several variables (or their equivalent).

Intended learning outcomes

 

To give deeper knowledge and improved skills in the subjects of ordinary differential equations (ODEs) and continuous-time dynamical systems. After completing the course, the student should be able to:

  • Use some classical methods of finding exact solutions of ODEs.
  • Qualitatively investigate dynamical systems (mainly in two dimensions): phase portrait, stability of equilibria.

  • Formulate and analyze ODEs using simple models from physics, chemistry and biology.

Course content

  • Some exact methods of solution for certain types of ODEs.
  • Existence and uniqueness theorems.
  • The fundamental matrix of a linear system of ODEs.
  • The matrix exponential function.
  • Continuous-time dynamical systems (systems of autonomous first-order ODEs).
  • Linear dynamical systems: exact solution, phase portrait, classification.
  • Nonlinear dynamical systems: phase portrait, linearization at equilibria, stability, limit cycles, bifurcations.
  • Various examples of applications.
  • Introduction to solving ODEs numerically and symbolically on the computer.
  • Introduction to Poincaré maps, discrete-time dynamical systems and chaos theory.

Teaching and working methods

Lectures, problem classes and computer exercises.

Examination

TEN1Written examinationU, 3, 4, 54 credits
UPG1AssignmentsU, G2 credits

Grades

Four-grade scale, LiU, U, 3, 4, 5

Department

Matematiska institutionen

Director of Studies or equivalent

Jesper Thorén

Examiner

Hans Lundmark

Education components

Preliminary scheduled hours: 44 h
Recommended self-study hours: 116 h

Course literature

Books
D. K. Arrowsmith & C. M. Place, (1992) Dynamical Systems: Differential Equations, Maps and Chaotic Behaviour Chapman & Hall/CRC
Other
Kurt Hansson, Föreläsningar om ordinära differentialekvationer (pdf-dokument)

Books

D. K. Arrowsmith & C. M. Place, (1992) Dynamical Systems: Differential Equations, Maps and Chaotic Behaviour Chapman & Hall/CRC

Other

Kurt Hansson, Föreläsningar om ordinära differentialekvationer (pdf-dokument)
TEN1 Written examination U, 3, 4, 5 4 credits
UPG1 Assignments U, G 2 credits

Course syllabus

A syllabus has been established for each course. The syllabus specifies the aim and contents of the course, and the prior knowledge that a student must have in order to be able to benefit from the course.

Timetabling

Courses are timetabled after a decision has been made for this course concerning its assignment to a timetable module. A central timetable is not drawn up for courses with fewer than five participants. Most project courses do not have a central timetable.

Interrupting a course

The vice-chancellor’s decision concerning regulations for registration, deregistration and reporting results (Dnr LiU-2015-01241) states that interruptions in study are to be recorded in Ladok. Thus, all students who do not participate in a course for which they have registered must record the interruption, such that the registration on the course can be removed. Deregistration from a course is carried out using a web-based form: www.lith.liu.se/for-studenter/kurskomplettering?l=sv. 

Cancelled courses

Courses with few participants (fewer than 10) may be cancelled or organised in a manner that differs from that stated in the course syllabus. The board of studies is to deliberate and decide whether a course is to be cancelled or changed from the course syllabus. 

Regulations relating to examinations and examiners 

Details are given in a decision in the university’s rule book: http://styrdokument.liu.se/Regelsamling/VisaBeslut/622678.

Forms of examination

Examination

Written and oral examinations are held at least three times a year: once immediately after the end of the course, once in August, and once (usually) in one of the re-examination periods. Examinations held at other times are to follow a decision of the board of studies.

Principles for examination scheduling for courses that follow the study periods:

  • courses given in VT1 are examined for the first time in March, with re-examination in June and August
  • courses given in VT2 are examined for the first time in May, with re-examination in August and October
  • courses given in HT1 are examined for the first time in October, with re-examination in January and August
  • courses given in HT2 are examined for the first time in January, with re-examination at Easter and in August.

The examination schedule is based on the structure of timetable modules, but there may be deviations from this, mainly in the case of courses that are studied and examined for several programmes and in lower grades (i.e. 1 and 2). 

  • Examinations for courses that the board of studies has decided are to be held in alternate years are held only three times during the year in which the course is given.
  • Examinations for courses that are cancelled or rescheduled such that they are not given in one or several years are held three times during the year that immediately follows the course, with examination scheduling that corresponds to the scheduling that was in force before the course was cancelled or rescheduled.
  • If teaching is no longer given for a course, three examination occurrences are held during the immediately subsequent year, while examinations are at the same time held for any replacement course that is given, or alternatively in association with other re-examination opportunities. Furthermore, an examination is held on one further occasion during the next subsequent year, unless the board of studies determines otherwise.
  • If a course is given during several periods of the year (for programmes, or on different occasions for different programmes) the board or boards of studies determine together the scheduling and frequency of re-examination occasions.

Registration for examination

In order to take an examination, a student must register in advance at the Student Portal during the registration period, which opens 30 days before the date of the examination and closes 10 days before it. Candidates are informed of the location of the examination by email, four days in advance. Students who have not registered for an examination run the risk of being refused admittance to the examination, if space is not available.

Symbols used in the examination registration system:

  ** denotes that the examination is being given for the penultimate time.

  * denotes that the examination is being given for the last time.

Code of conduct for students during examinations

Details are given in a decision in the university’s rule book: http://styrdokument.liu.se/Regelsamling/VisaBeslut/622682.

Retakes for higher grade

Students at the Institute of Technology at LiU have the right to retake written examinations and computer-based examinations in an attempt to achieve a higher grade. This is valid for all examination components with code “TEN” and "DAT". The same right may not be exercised for other examination components, unless otherwise specified in the course syllabus.

Retakes of other forms of examination

Regulations concerning retakes of other forms of examination than written examinations and computer-based examinations are given in the LiU regulations for examinations and examiners, http://styrdokument.liu.se/Regelsamling/VisaBeslut/622678.

Plagiarism

For examinations that involve the writing of reports, in cases in which it can be assumed that the student has had access to other sources (such as during project work, writing essays, etc.), the material submitted must be prepared in accordance with principles for acceptable practice when referring to sources (references or quotations for which the source is specified) when the text, images, ideas, data, etc. of other people are used. It is also to be made clear whether the author has reused his or her own text, images, ideas, data, etc. from previous examinations.

A failure to specify such sources may be regarded as attempted deception during examination.

Attempts to cheat

In the event of a suspected attempt by a student to cheat during an examination, or when study performance is to be assessed as specified in Chapter 10 of the Higher Education Ordinance, the examiner is to report this to the disciplinary board of the university. Possible consequences for the student are suspension from study and a formal warning. More information is available at https://www.student.liu.se/studenttjanster/lagar-regler-rattigheter?l=sv.

Grades

The grades that are preferably to be used are Fail (U), Pass (3), Pass not without distinction (4) and Pass with distinction (5). Courses under the auspices of the faculty board of the Faculty of Science and Engineering (Institute of Technology) are to be given special attention in this regard.

  1. Grades U, 3, 4, 5 are to be awarded for courses that have written examinations.
  2. Grades Fail (U) and Pass (G) may be awarded for courses with a large degree of practical components such as laboratory work, project work and group work.

Examination components

  1. Grades U, 3, 4, 5 are to be awarded for written examinations (TEN).
  2. Grades Fail (U) and Pass (G) are to be used for undergraduate projects and other independent work.
  3. Examination components for which the grades Fail (U) and Pass (G) may be awarded are laboratory work (LAB), project work (PRA), preparatory written examination (KTR), oral examination (MUN), computer-based examination (DAT), home assignment (HEM), and assignment (UPG).
  4. Students receive grades either Fail (U) or Pass (G) for other examination components in which the examination criteria are satisfied principally through active attendance such as other examination (ANN), tutorial group (BAS) or examination item (MOM).

The examination results for a student are reported at the relevant department.

Regulations (apply to LiU in its entirety)

The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.

LiU’s rule book for education at first-cycle and second-cycle levels is available at http://styrdokument.liu.se/Regelsamling/Innehall/Utbildning_pa_grund-_och_avancerad_niva. 

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