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Dynamical Systems Analysis (Mandatory in MathBio / Elective in PSM / Advanced Mathematics)

Title
Dynamical Systems Analysis (Mandatory in MathBio / Elective in PSM / Advanced Mathematics)
Semester
F2023
Master programme in
Mathematics * / Mathematical Physical Modelling * / Mathematical Computer Modelling * / Mathematical Bioscience * / Mathematical Bioscience / Physics and Scientific Modelling
Type of activity

Course

Teaching language
English
Study regulation

Read about the Master Programme and find the Study Regulations at ruc.dk

Læs mere om uddannelsen og find din studieordning på ruc.dk

REGISTRATION AND STUDY ADMINISTRATIVE
Registration

Sign up for study activities at stads selvbetjening within the announced registration period, as you can see on the Studyadministration homepage.

When signing up for study activities, please be aware of potential conflicts between study activities or exam dates.

The planning of activities at Roskilde University is based on the recommended study programs which do not overlap. However, if you choose optional courses and/or study plans that goes beyond the recommended study programs, an overlap of lectures or exam dates may occur depending on which courses you choose.

Number of participants
ECTS
5
Responsible for the activity
Morten Andersen (moan@ruc.dk)
Head of study
Jesper Schmidt Hansen (jschmidt@ruc.dk)
Teachers
Study administration
INM Registration & Exams (inm-exams@ruc.dk)
Exam code(s)
U60165 / U41331 / U41560
ACADEMIC CONTENT
Overall objective

The overall objective of the course is to give the student an advanced understanding dynamical systems and how analysis of these are constructed.

Detailed description of content

The course includes examples of dynamical systems (ordinary differential equartions) that arise in physics, chemistry and biology. The focus is on the mathematical analysis and mathematical properties of such dynamical systems.

Systems of linear differential equations with constant coefficients are covered in detail, showing important applications of linear algebra. Methodology from analysis I and II is then used extensively for nonlinear systems of differential equations, leading to the proof of the contraction mapping theorem on Banach spaces which is used to prove the existence and uniqueness theorem. The focus is then on the qualitative behaviour of solution trajectories, involving Lyapunov stability and attractors.

The expected outcome for the student is a solid mathematical understanding of dynamical systems and their qualitative properties, including stating, proving and contextualize theorems of dynamical systems

Course material and Reading list

The course will have a primary textbook which will be announced on moodle prior to semesterstart.

Overall plan and expected work effort

The teaching format is based on a scientific dialogue between the students and the course teacher, working with exercises, student presentations, etc. The teacher will, of course, highlight relevant points. For the dialogue to be fruitful, the student must prepare for each class; this includes careful reading the text material, finish exercises, and other home work suggested by the teacher. As a rule of thumb, the student should use 1-2 hours of preparation for every hour in class.

  • Total (minimum): 135 hours

  • In class: 40 hours

  • Preparation for class: 80 hours

  • Preparation for exam: 15 hours

Format
Evaluation and feedback

The course includes formative evaluation based on dialogue between the students and the teacher(s).

Students are expected to provide constructive critique, feedback and viewpoints during the course if it is needed for the course to have better quality. Every other year at the end of the course, there will also be an evaluation through a questionnaire in SurveyXact. The Study Board will handle all evaluations along with any comments from the course responsible teacher.

Furthermore, students can, in accordance with RUCs ‘feel free to state your views’ strategy through their representatives at the study board, send evaluations, comments or insights form the course to the study board during or after the course.

Programme

Depending on the specific topic, the teacher, and the student group, the students will engage in a dialogue with the teacher and from this do exercises in groups or individually. The exercises will be based on pure mathematical analysis, computer-aided analysis, discussion in groups, with teacher, and so forth.

The themes in this course are:

Linear differential equations with constant coefficients, the matrix exponential, phase plane analysis, the contraction mapping theorem, existence and uniqueness theorem of nonlinear ordinary differential equations, definition of flow, Lyapunov stability of equilibria, attractors, applications within physics, biology and chemistry.

ASSESSMENT
Overall learning outcomes

After the course the student will be able to

  • formulate mathematical analysis of non-linear differential equation systems, e.g., via phase plane analysis.

  • perform local and global stability analysis.

  • demonstrate in-depth knowledge about bifurcations and how these affect the dynamics in mathematical models.

  • present results from the mathematical analysis in a clear and concise manner using mathematical formalism and reasoning.

  • critically assess the mathematical methodology behind analysis of dynamical systems analysis

Form of examination

Individual oral exam based on a portfolio.

The character limit of the portfolio is 1,200-120,000 characters, including spaces. Examples of written products are exercise responses, talking points for presentations, written feedback, reflections, written assignments. The preparation of the products may be subject to time limits.
The character limits include the cover, table of contents, bibliography, figures and other illustrations, but exclude any appendices.

Time allowed for exam including time used for assessment: 30 minutes.
The assessment is an assessment of the oral examination. The written product(s) is not part of the assessment.

Permitted support and preparation materials for the oral exam: All.

Assessment: 7-point grading scale.
Moderation: Internal co-assessor
Form of Re-examination
Samme som ordinær eksamen / same form as ordinary exam
Type of examination in special cases
Examination and assessment criteria

Individual oral exam based on a portfolio constructed from a mini project and working with known exam questions during the course to build a portfolio for the oral exam.

The student begin the exam with a presentation, after the presentation there will be a dialogue between the student, assessor and co-assessor.

The assessment criteria of the written part

  • formulate mathematical analysis of non-linear differential equation systems, e.g., via phase plane analysis.

  • perform local and global stability analysis.

  • demonstrate in-depth knowledge about bifurcations and how these affect the dynamics in mathematical models.

  • present results from the mathematical analysis in a clear and concise manner using mathematical formalism and reasoning.

  • critically assess the mathematical methodology behind analysis of dynamical systems analysis

The assessment of the oral exam is based on the student’s ability to meet the criteria mentioned above and their ability to

  • clearly present and communicate the scientific content of the portfolio

  • engage in a scientific dialogue and discussion with the assessor and co assessor

Furthermore, whether the performance meets all formal requirements in regard to both for the written og oral exam

Exam code(s)
Exam code(s) : U60165 / U41331 / U41560
Last changed 06/03/2023

lecture list:

Show lessons for Subclass: 1 Find calendar (1) PDF for print (1)

Tuesday 07-02-2023 08:15 - 07-02-2023 12:00 in week 06
Dynamical Systems Analysis (MATHBIO)

Tuesday 14-02-2023 08:15 - 14-02-2023 12:00 in week 07
Dynamical Systems Analysis (MATHBIO)

Tuesday 21-02-2023 08:15 - 21-02-2023 12:00 in week 08
Dynamical Systems Analysis (MATHBIO)

Tuesday 28-02-2023 08:15 - 28-02-2023 12:00 in week 09
Dynamical Systems Analysis (MATHBIO)

Tuesday 07-03-2023 08:15 - 07-03-2023 12:00 in week 10
Dynamical Systems Analysis (MATHBIO)

Tuesday 14-03-2023 08:15 - 14-03-2023 12:00 in week 11
Dynamical Systems Analysis (MATHBIO)

Tuesday 21-03-2023 08:15 - 21-03-2023 12:00 in week 12
Dynamical Systems Analysis (MATHBIO)

Tuesday 28-03-2023 08:15 - 28-03-2023 12:00 in week 13
Dynamical Systems Analysis (MATHBIO)

Tuesday 11-04-2023 08:15 - 11-04-2023 12:00 in week 15
Dynamical Systems Analysis (MATHBIO)

Tuesday 18-04-2023 08:15 - 18-04-2023 12:00 in week 16
Dynamical Systems Analysis (MATHBIO)

Tuesday 25-04-2023 08:15 - 25-04-2023 12:00 in week 17
Dynamical Systems Analysis (MATHBIO)

Thursday 25-05-2023 10:00 - 25-05-2023 10:00 in week 21
Dynamical Systems Analysis - Hand-in of portfolio (MathBio)

Thursday 01-06-2023 08:15 - 01-06-2023 16:00 in week 22
Dynamical Systems Analysis - Exam (MATHBIO)

Friday 30-06-2023 10:00 - 30-06-2023 10:00 in week 26
Dynamical Systems Analysis - Hand-in of portfolio (reexam) (MathBio)

Monday 14-08-2023 08:15 - 14-08-2023 16:00 in week 33
Dynamical Systems Analysis - Reexam (MATHBIO)