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. 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.

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