# Dynamics of complex systems

## Olat Link

Science likes to deal with simple systems that consist of only a few components and are governed by simple, linear laws. Because such systems are easy to analyze and understand.

Most real systems, however, consist of many components, and the laws between these components are non-linear (in simple terms: "twice as much in = twice as much out" does not apply). Such systems are called complex. They develop their own unique dynamics, which at first glance appear confusing and disorderly, but nevertheless exhibit a number of interesting laws. The brain is such a system, but also population dynamics, social interactions or economic processes can be understood as complex dynamic systems.

In this seminar we will first deal with the terms fractal and self-similarity. This is because complex systems often show self-similar structures with fractal dimensions in their dynamics. Thereby we will learn about Koch curves (see above), Mandelbrot sets and Mandelbrot-Set (see right). Then we study a very simple nonlinear system that leads to the well-known fig tree chaos. The concept of bifurcation is explained and a new natural constant, the fig tree constant, is introduced. With this and other chaotic systems (three-body problem) it becomes clear that even in a deterministic system, the smallest changes can cause arbitrarily large changes in the long run: the well-known "butterfly effect" (the wing beat of a butterfly in Brazil determines whether a tornado will break out in Texas; Lorenz, 1972). We get to know the term strange attractor. - And what does this have to do with the brain? The brain is a massively nonlinear system, both in its overall behavior and in its components (neurons, synapses, down to the behavior of individual ion channels and proteins). The dynamics of chaotic systems can therefore be used as a metaphor for the dynamics of the brain.

Chaos can already be studied in systems with only a few components. The brain has 1010 neurons. What is qualitatively new when we look at systems with many components involved? First, we study so-called emergent properties, which cannot be predicted from microscopic rules, on cellular automata (artificial life). Then we turn to a model that was originally designed to describe magneticity. However, the so-called spin-glasses were soon recognized as a simple model for the activity of neural networks (Hopfield networks). Here, too, we study emergent properties; these include phase transitions and critical points, as they also play a role in brain dynamics (sleep/wake phases, epilepsy). The dynamics of Hopfield nets are compared with memory dynamics. Finally, we drop the limitations of the Hopfield network and study more realistic neural networks that can no longer be described simply by an energy function. In the process, phenomena of self-organization occur, as they can also be observed in neural maps.

Knowledge in linear algebra and analysis is neither assumed nor taught - the goal of the seminar is not a deeper understanding of the underlying mathematics. The goal of the seminar is not a deeper understanding of the underlying mathematics, but rather to acquire a basis for a non-mathematical, but nevertheless solid intuition about the dynamics of complex systems by means of examples reproduced on the computer itself in the sense of interactive hands-on didactics. The participants should be able to distinguish between serious applications of the theory of complex systems and dubious chaos metaphors. The demonstrations and exercises will be performed in Python. Programming knowledge is not required, but the willingness to acquire it is. For further support there will be tutorials.

- Adam, Stefan, Matlab und Mathematik kompetent einsetzen. Eine Einführung für Ingenieure und Naturwissenschaftler.Wiley-VCH Verlag GmbH, 2006. Kapitel 1
- Peitgen, H.-O., Jürgens, H., Saupe, D., Bausteine des Chaos: Fraktale. rororo 1998.
- Peitgen, H.-O., Jürgens, H., Saupe, D., Chaos: Bausteine der Ordnung. rororo 1998.
- Schmidhuber, Christof: Der Phasenübergang zwischen Wasser und Dampf