Chaos
Kaos

FMFN05F, 7.5 credits

Valid from: Spring 2019
Decided by: Anders Gustafsson FTF-AGU
Date of establishment: 2019-05-26

General Information

Division: Mathematical Physics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: FMFN05
Teaching language: English

Aim

The course aims at giving an introduction to chaotic systems, i.e. non-linear systems that are deterministic but with a time development which is not predictable over longer periods. The course should give a possibility to reflect over the fascinating phenomena which may show up in chaotic systems, e.g. strange attractors and in this context a basic comprehension of the importance of fractal geometry, or the posibility that the solar system is instable over a longer time scale.

Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• have a general knowledge about system conditions leading to chaotic and regular behaviour, respectively.
• be familiar with mathematical methods used to analyse chaotic systems.
• have a general understanding why it is useful to introduce dimensions which are not integer.

Competences and Skills

For a passing grade the doctoral student must

• be able to apply mathematical methods used for the description of non-linear systems.
• be able to analyse the time development of a system and be able to determine if the system is chaotic or regular.
• be able to determine which mathematical models are appropriate in different situations.
• be able to determine the dimension of simple fractals.

Course Contents

Temporally discrete systems. Feigenbaum’s theory of branching. Dependence on initial values. Fractal geometry with various applications. Different definitons of dimensions Dissipative systems. Systems of differential equations. Phase space and the Poincaré section. Lyapunov exponents and strange attractors. Coupled oscillators and frequency locking. Conservative systems and the KAM theory. Hamilton's formalism, integrable systems, billiards, area-preserving maps, chaotic motion in the solar system.

Course Literature

• Ohlén, G., Åberg, S. & Östborn, P.: Chaos, Compendium. Lund 2006.. Lund, 2006.
• Strogatz, S. H.: Nonlinear dynamics and Chaos. Westview Press. ISBN 9780813349107.

Instruction Details

Types of instruction: Lectures, laboratory exercises, project

Examination Details

Examination formats: Written exam, written report. Demonstrate competences in a written exam and presentation of a project. Compulsory computer exercise. The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.
Grading scale: Failed, pass
Examiner:

Admission Details

Assumed prior knowledge: Elementary mathematics and mechanics. Multivariate calculus and elementary partial differential equations.

Course Occasion Information

Contact and Other Information

Course coordinators:
Web page: http://www.matfys.lth.se/education/FMFN05/