CHAOS FOR SCIENCE AND TECHNOLOGY | FMF090 |

**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.

*Knowledge and understanding*

For a passing grade the 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

*Skills and abilities*

For a passing grade the 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

**Contents***Temporally discrete systems.* Feigenbaums 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.

**Literature**

Ohlén, G, Åberg, S, Östborn, P: Chaos, Compendium. Lund 2002