Chaos and Non-Linear Phenomena, 6 credits (TFYA36)

Kaos och icke-linjära fenomen, 6 hp

Main field of study

Mathematics Applied Mathematics Applied Physics Physics

Level

Second cycle

Course type

Programme course

Examiner

Magnus Johansson

Director of studies or equivalent

Magnus Johansson

Available for exchange students

Yes
Course offered for Semester Period Timetable module Language Campus VOF
6KFYN 6 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYY Applied Physics and Electrical Engineering, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYY Applied Physics and Electrical Engineering, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6MFYS Physics and Nanoscience, Master's programme 2 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering (Theory, Modelling and Visualization) 8 (Spring 2017) 1, 2 3, 2 Swedish/English Linköping v

Main field of study

Mathematics, Applied Mathematics, Applied Physics, Physics

Course level

Second cycle

Advancement level

A1X

Course offered for

  • Applied Physics and Electrical Engineering, M Sc in Engineering
  • Physics and Nanoscience, Master's programme
  • Applied Physics and Electrical Engineering - International, M Sc in Engineering

Specific information

The course is offered every second year. It will not be available during 2017.

Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.

Prerequisites

Linear algebra, Modern Physics, Classical Mechanics and some knowledge of Fourier Analysis and Partial Differential Equations. Analytical Mechanics, Quantum Mechanics and Physics of Condensed Matter are recommended.

Intended learning outcomes

The aim of the course is that the students should get an orientation in the special properties of non-linear systems of theoretical and practical interest, and also an orientation in some topical areas of research.

After finishing the course the student knows how to:

  • explain basic concepts in non-linear physics such as chaos, fractal dimensions, integrability, solitons and quantum chaos, and exemplify important applications of these concepts in different scientific disciplines.
  • model simple non-linear dynamical systems mathematically, and demonstrate some of their characteristic properties by analyzing bifurcations and calculating Lyapunov-exponents and fractal dimensions for their attractors.
  • explain mechanisms for appearance of chaos in conservative systems and properties of the corresponding quantum mechanical systems, and solve problems by applying the theory for Hamiltonian systems on simple models.
  • derive soliton solutions in special mathematical models for wave motion in non-linear media, and exemplify their physical significance.

Course content

Introduction. Experiments and simple models. Vibrations in mechanical systems and electrical circuits. Piecewise linear maps and deterministic chaos. Universal behavior of quadratic maps. Bifurcations. Poincaré maps. Lyapunov exponents. Fractal dimensions. Period-doubling. Homoclinic and heteroclinic orbits. The intermittency route to chaos. Strange attractors in dissipative dynamical systems. The transition from quasiperiodicity to chaos. Regular and irregular motion of conservative systems. Integrable and non-integrable systems. Solitons and breathers with applications. Chaos in quantum systems.

Teaching and working methods

The course consists of lectures and exercises solving problems.

Examination

UPG1Hand-in assignmentsU, 3, 4, 55.5 credits
UPG2Oral Presentation of Solved ProblemsU, G0.5 credits

Grades

Four-grade scale, LiU, U, 3, 4, 5

Department

Institutionen för fysik, kemi och biologi

Director of Studies or equivalent

Magnus Johansson

Examiner

Magnus Johansson

Course website and other links

http://www.ifm.liu.se/courses/tfya36/kaos.html

Education components

Preliminary scheduled hours: 56 h
Recommended self-study hours: 104 h

Course literature

Additional literature

Compendiums
G. Ohlén/S. Åberg/P. Östborn, Chaos

Compendium, Lund

Other

Complementary material handed out during the course. 

Additional literature

Compendia

G. Ohlén/S. Åberg/P. Östborn, Chaos

Compendium, Lund

Other

Complementary material handed out during the course. 

UPG1 Hand-in assignments U, 3, 4, 5 5.5 credits
UPG2 Oral Presentation of Solved Problems U, G 0.5 credits

Regulations (apply to LiU in its entirety)

The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.

LiU’s rule book for education at first-cycle and second-cycle levels is available at http://styrdokument.liu.se/Regelsamling/Innehall/Utbildning_pa_grund-_och_avancerad_niva. 

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