## 3.7 Strange Attractors

The long-term behavior of a system can be represented in the so-called n-dimensional “phase space”. There the specific time development is described through a trajectory, through which the history of the system is recorded. Attraction areas, to which these trajectories aim, are called attractors. Put into other words an attractor is a preferred position for the system to which it evolves no matter what the starting position is. Once such a position is reached it will then stay on the attractor in the absence of other factors.

The existence of an attractor[01] in general means for a scientific process that it possesses the characteristic either to run in a stable, periodical or quasi-periodical way. ‘Stable’ means that the system aims at a certain end condition, called point attractor or the fixed point of the system. On the other hand a process is ‘periodical’ if it repeats itself through a certain interval of time. Finally ‘quasi-periodical’ means that it lasts some time at the beginning until it turns into a periodical behavior - see picture 21.

picture 21: Strange attractor

In the chapter about non-linear, chaotic fractals we looked at what is happening to a certain point on the screen and depending on its behavior we colored it differently. Now we follow the iteration of the number instead. The number is used in a formula and the new number resulting from that is fixed on the screen, then this new number is again used in the same formula and fixed on the screen, and so on - this technique is also-called "hop-along".

One classical strange attractor is the “Lorenz attractor” that is used for the weather forecast. The weather forecast depends on many parameters such as season, vegetation, temperature or direction of the wind. Edward Lorenz tried to describe meteorological processes with the support of differential equations - e.g. the model describing the earth’s atmosphere has the form of

[http://stud4.tuwien.ac.at/~e9426503/soinfoges/mathematik.html (24.09.2001).]

The “Lorenz attractor” consists of one continuous infinitely long curve. Following a point on it will indicate that no curve is passed through twice, which means that the system behaves chaotic because nothing is regular. But nevertheless the function does not pass over certain borders in its long-term behavior. Zooming into the "Lorenz attractor" the line is split up and we will always find new structures. So it is not possible to locate exactly where the system is at a certain moment. From that follows that the weather can only be forecast for a short period.[Voß Herbert, Chaos und Fraktale - selbst programmieren (1994), Franzis-Verlag GmbH Österreich, ISBN 3-7723-7003-9, p.43.]

Another category of attractors is called strange attractors, whose name arises from their strange characteristics. They consist of an infinite sequence and offer an unpredictable chaotic behavior, but nevertheless in the phase space they occupy a sub-room of lower dimension. Looking at neighboring trajectories their expansions follow completely different directions. From that follows that though the system evolves to and remains on the attractor, it is not possible to give a long-term behavior - see picture 21. This category is often applied to represent chaotic systems[02] .

## Footnotes

[01] The word “attractor” originates from the Latin “attrahere”. Voß Herbert, Chaos und Fraktale - selbst programmieren (1994), Franzis-Verlag GmbH Österreich, ISBN 3-7723-7003-9, p.26.
[02] In the geometrical expression a strange attractor is a fractal.

DAtTE - Detection of Attic Extensions: Workflow to analyze the potentials of roofs in an urban environment
Talk and Proceeding: eCAADe 2021 - Towards a new, configurable architecture (Novi Sad, Serbia, 2021 | hybrid conference)
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