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  
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 Studieadministration (inmstudieadministration@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 12 hours of preparation for every hour in class.

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, computeraided 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

Form of examination 
Individual oral exam based on a portfolio. The character limit of the portfolio is 1,200120,000 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: 7point grading scale. Moderation: Internal coassessor 
Form of Reexamination 
Samme som ordinær eksamen

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 coassessor. The assessment criteria of the written part
The assessment of the oral exam is based on the student’s ability to meet the criteria mentioned above and their ability to
Furthermore, whether the performance meets all formal requirements in regard to both for the written og oral exam 
Exam code(s)  
Last changed  19/09/2022 