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Syllabus for

Academic year
TIF155 - Dynamical systems
Dynamiska system
 
Syllabus adopted 2020-02-12 by Head of Programme (or corresponding)
Owner: MPCAS
7,5 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: Second-cycle
Major subject: Engineering Physics
Department: 16 - PHYSICS


Teaching language: English
Application code: 11112
Open for exchange students: Yes
Block schedule: B

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0107 Examination 7,5c Grading: TH   7,5c   11 Jan 2021 am J   09 Apr 2021 pm J,  18 Aug 2021 am J

In programs

MPCAS COMPLEX ADAPTIVE SYSTEMS, MSC PROGR, Year 1 (compulsory)
MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 2 (elective)
MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 1 (compulsory elective)
MPSYS SYSTEMS, CONTROL AND MECHATRONICS, MSC PROGR, Year 1 (elective)

Examiner:

Kristian Gustafsson

  Go to Course Homepage


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 give an understanding of theoretical concepts and practical aspects arising in the description of nonlinear dynamical systems: how is chaos measured and characterised? How can one detect deterministic chaos in an experimental time series? How can one control and predict chaotic systems? Applications in physics, biology, and economics are described.

Learning outcomes (after completion of the course the student should be able to)

After successfully completing this course the students shall 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

Regular dynamics:
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.

Chaotic dynamics:
Lyapunov exponents.
Strange attractors.
Fractal dimension, fractals in physical systems.
Transitions to chaos.

Chaos and regular dynamics in Hamiltonian systems.

Organisation

Lectures, set 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 four sets of homework assignments (50%) and a written examination (50%).


Published: Mon 28 Nov 2016.