The aim of this course is to give an understanding of fundamental concepts used to describe complex systems. As examples, chaotic low-dimensional systems, self-organizing systems, and cellular automata are discussed.

The fact that systems composed of a large number of simple components can exhibit complex phenomena is exemplified in, for example, self-organising systems (in the form of chemical reaction-diffusion systems), the second law of thermodynamics (as a statistical result of large physical systems), neural networks, evolution of cooperation, cellular automata (as an example of an abstract computational class of systems), economic systems of interacting trading agents, urban growth and traffic systems.

There is no universal definition of a complex system, but there are several features that are usually brought up when a system is considered complex, typically involving order/disorder and correlations. An important scientific question is whether these and other characteristics of the systems can be quantified in a comprehensive way. This is one of the aims with this course -- to provide a set of tools that can be used to give a quantitative description of a complex system for a variety of different types of systems. All quantities derived can be interpreted in terms of information, referring to the information quantity introduced by Shannon and Weaver.

Information-theoretic concepts can be applied on the macro-level of a system, for example, in order to describe the spatial structure formed in a chemical self-organizing system. The connection between information theory and statistical mechanics makes it possible to relate such an analysis to the thermodynamic properties and limitations of the system. Other examples include complex phenomena in cellular automata, entropy in microscopic physical systems (spin systems), and chaotic behaviour in dynamical systems.

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