1. Introduction

There exist several different possibilities how one can visualize dynamical systems. One difference is in whether one visualizes local or global characteristics. Another difference can be found in abstract versus direct visualization of the features of the system.

In my project, the main topic was the visualization for fixed points. Fixed points are, as the name already tells, points in the phase space, which you can't leave, once you have reached them. They don't change over time, unlike all other states in the space. These fixed points can have different characteristics as to how states in their near neighbourhood are influenced by them. They can be, for example attractive, meaning that all states in a small enough neighbourship around them are attracted by them and starting in one of these states means, that you will reach the fixed point after (infinite) long time has gone by. They can also be reppeling (that describes the inverse behavior of the above described), or they can be saddles (meaning that there exist directions from which you are attracted towards the fixed point, and inbetween there are directions, where you are repelled from it), or they can have even more strange characteristics as to how they influence their suroundings.

We used several different approaches to the problems, that arise, when trying to visualize the characteristics of the different fixed points of dynamical systems, and these approaches as well as all the solutions we found to be suitable will be investigated and presented in section 2 of this document. In section 3 other visualization techniques for dynamical systems, not concerning the special case of fixed points visualization but more general, will be shortly presented. These were also developed during this project, as well as some tools necessary for all these visualization techniques. The tools will be presented in section 4, and in section 5 the most interesting results will be shown.