Kaos

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

**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

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.

*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.

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.

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

**Types of instruction:** Lectures, laboratory exercises, project

**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:**

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

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