Course syllabus adopted 2024-02-07 by Head of Programme (or corresponding).
Overview
- Swedish nameDynamiska system
- CodeTIF155
- Credits7.5 Credits
- OwnerMPCAS
- Education cycleSecond-cycle
- Main field of studyEngineering Physics
- DepartmentPHYSICS
- GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Course round 1
- Teaching language English
- Application code 11118
- Block schedule
- Open for exchange studentsYes
Credit distribution
Module | Sp1 | Sp2 | Sp3 | Sp4 | Summer | Not Sp | Examination dates |
|---|---|---|---|---|---|---|---|
| 0107 Examination 7.5 c Grading: TH | 7.5 c |
|
In programmes
- MPCAS - COMPLEX ADAPTIVE SYSTEMS, MSC PROGR, Year 1 (compulsory)
- MPENM - ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 1 (compulsory elective)
- MPENM - ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 2 (elective)
- MPSYS - SYSTEMS, CONTROL AND MECHATRONICS, MSC PROGR, Year 1 (elective)
Examiner
- Kristian Gustafsson
- Senior Lecturer, Institution of physics at Gothenburg University
Eligibility
General entry requirements for Master's level (second cycle)Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.
Specific entry requirements
English 6 (or by other approved means with the equivalent proficiency level)Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.
Course specific prerequisites
Sufficient knowledge of Mathematics (analysis in one real variable, linear algebra), basic programming skills.Aim
The aim of the course is to provide students with a comprehensive understanding of both theoretical concepts and practical aspects arising in the analysis of nonlinear dynamical systems. The primary focus lies on investigating the various possible long-term solutions of such systems, and how these change when the parameters of the system change. Special attention is dedicated to chaotic solutions through the introduction of methods to detect chaotic dynamics and how it can be characterized. Throughout the course, students will gain insights into the practical utility of dynamical systems in different areas such as physics, biology, and economics.Learning outcomes (after completion of the course the student should be able to)
- understand and explain key concepts in regular dynamical systems
- perform linear stability analysis, and understand its limitations
- analyze qualitative changes in the system as control parameters change (bifurcations)
- understand and explain the key concepts used in describing deterministic chaos in non-linear systems
- efficiently simulate dynamical systems on a computer
- numerically compute Lyapunov exponents and fractal dimensions
- efficiently search for periodic orbits and determine their stabilities
- recognize and analyse chaotic dynamics in initially unfamiliar contexts
- present numerical results graphically in a clear and concise manner
- communicate results and conclusions in a clear and logical fashion
Content
- Continuous flows
- Fixed points and stability analysis
- Characterisation of linear and non-linear flows
- Bifurcations och structural stability
- Index theory
- Periodic motion, limit cycles and relaxation oscillators
- Lyapunov exponents
- Strange attractors
- Fractal dimension, fractals in physical systems
- Transitions to chaos
Organisation
Lectures, sets of homework problems, examples classes, and written exam.
Literature
Lecture notes will be made available.
Course book: Nonlinear Dynamics and Chaos, by Stephen H. Strogatz.
Recommended additional material:
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by Guckenheimer and Holmes
ChaosBook by Cvitanovic
Examination including compulsory elements
The final grade is based on homework assignments (50%) and a written examination (50%).
The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers on educational support due to disability.