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Graduate courses

Departments' graduate courses for PhD-students.

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Syllabus for

Academic year
TIF155 - Dynamical systems
 
Syllabus adopted 2015-02-20 by Head of Programme (or corresponding)
Owner: MPCAS
7,5 Credits
Grading: TH - Five, Four, Three, Not passed
Education cycle: Second-cycle
Major subject: Engineering Physics
Department: 16 - PHYSICS


Teaching language: English
Open for exchange students
Block schedule: B

Course module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0107 Written and oral assignments 7,5 c Grading: TH   7,5 c    

In programs

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

Examiner:

Professor  Stellan Östlund



Eligibility:


In order to be eligible for a second cycle course the applicant needs to fulfil the general and specific entry requirements of the programme that owns the course. (If the second cycle course is owned by a first cycle programme, second cycle entry requirements apply.)
Exemption from the eligibility requirement: Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling these requirements.

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 the key concepts used in describing deterministic chaos in non-linear systems; 
efficiently simulate dynamical systems on a computer;
numerically compute Lyapunov exponents; efficiently search for periodic orbits anddetermine their stabilities; recognize and analyse chaotic dynamics in initially unfamiliar contexts and in other disciplines (for example in medicine, biology, or in the engineering sciences); 
write well-structured technical reports in English presenting and explaining analytical calculations and numerical results
communicate results and conclusions in a clear and logical fashion.

Content

One-dimensional iterated maps: periodic orbits, stability, symbolic dynamics, natural invariant density, Perron-Frobenius operator. Transition to chaos: 
Lyapunov exponents, ergodicity
Bifurcations, structural stability.
Fractal dimension, fractals in physical systems. 
Chaotic dynamics: surface-of-section, invariant manifolds, Smale s horseshoe, hyperbolic sets, shadowing, Kolmogorov-Sinai entropy
Chaos and regular dynamics in Hamiltonian systems
Control of chaos
Chaos in high-dimensional systems, turbulence

Course home page

Organisation

Lectures, set homework problems, examples classes.
Web-based course evaluation.

Literature

Lecture notes will be made available.

Course book: Nonlinear Dynamics and Chaos, by Stephen H. Strogatz
.

Recommended additional material: Chaos in dynamical systems, E. Ott, Cambridge University Press, Cambridge 1993 (reprinted with corrections 1993, 1997).
Regular and Stochastic Motion, A. J. Lichtenberg and M. A. Lieberman, Springer-Verlag New York 1983; Differential equations, dynamical systems, and linear algebra, W. Hirsch and A. Smale, Academic Press, New York 1974

Examination

The final grade is based on homework assignments and in som cases by oral exam if this is agreed upon by the student and examiner. The examiner must be informed within a week after the course starts if a student would like to receive ECTS grades.


Page manager Published: Thu 04 Feb 2021.