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Syllabus for

Academic year
TIF155 - Dynamical systems
 
Syllabus adopted 2012-02-22 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)
MPSYS SYSTEMS, CONTROL AND MECHATRONICS, MSC PROGR, Year 1 (elective)
MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 2 (elective)

Examiner:

Professor  Stellan Östlund


Course evaluation:

http://document.chalmers.se/doc/91f7c6fa-085f-4436-b92c-ea6fdff6a015


Eligibility:

For single subject courses within Chalmers programmes the same eligibility requirements apply, as to the programme(s) that the course is part of.

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 and determine their stabilities
recognize and analyse chaotic dynamics in initially unfamiliar contexts, 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 scattering
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: Chaos in dynamical systems, E. Ott, Cambridge University Press, Cambridge 1993 (reprinted with corrections 1993, 1997).

Recommended additional material:
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

Random matrices and chaotic dynamics, O. Bohigas, in: Chaos and quantum physics, Les Houches 1989, eds.: M.-J. Giannoni, A. Voros, and J. .Zinn-Justin, North-Holland, Amsterdam 1991

Examination

The examination is based on exercises and homework assignments (100%). The examiner must be informed within a week after the course starts if a student would like to receive ECTS grades.


Page manager Published: Mon 28 Nov 2016.