Attractors

An attractor is a set of states to which a complex system is attracted. The attraction comes from the complex interactions within the system itself, so an attractor is not like a magnet attracting metal, because that is something coming from the outside affecting the system. An attractor is generated within the system itself. An attractor has limit values, which define the boundary within which the values must land to be in the attractor, although the boundary marks are more fuzzy than fixed and definite.

There are several types of attractors. The first is a point attractor, where there is only one outcome for the system. Death is a point attractor for human beings. No matter who we are, how we lived our life or whatever, we die at the end of our life.

A point attractor on the phase space will be a basin. If a small ball is placed anywhere near the basin it will roll down to the same point at the bottom of the basin. There is no other place the ball could roll to.

The second type of attractor is called a limit cycle or periodic attractor. Instead of moving to a single state as in a point attractor, the system settles into a cycle. While we can then not predict the exact state of the system at any time, we know it will be somewhere in the cycle.

An example of a limit cycle is a predator-prey system. Imagine a lake with trout and a smaller number of pike. Those pike eat the young trout. Because there is so much food the number of pike increases. This increase in pike and all the young trout they eat means that the number of trout decreases. The drop in trout numbers means that the pike have less to eat and the pike numbers then decrease. This allows the trout numbers to increase again and the cycle begins over. The populations of trout and pike rise and fall in a cyclic fashion.

On the fitness landscape a limit cycle would be seen as a valley that goes round in a circle like the rim of a hat. If the small ball was placed anywhere near the attractor on the phase space, it would roll down into some point in the round track in the bottom of the valley and roll around the valley. We cannot tell which part it will roll to, but it will roll to a point on the limit cycle.

We can chart the progress of a limit cycle on a piece of paper as a circle or any shape that returns to itself to form a closed look. A more complicated limit cycle sweeps out a torus. A torus is shaped like a doughnut, so it exists in three dimensions not two. A double pendulum with one added on the end of the other when charted forms a torus shaped cycle.

The fourth type of attractor is called a strange attractor or a chaotic attractor. A strange attractor never repeats itself (or it would be periodic attractor), but the values always move towards a certain range of values. There are certain states in which the system can exist and others it cannot. If the system were to somehow move out from the acceptable range of states it would be “attracted” back into the attractor.

The giant red spot on Jupiter is a good example of a strange attractor. We know it has been stable since first viewed around 1660. The surface of Jupiter consists of belts of highly volatile gasses rotating around the planet at extremely high speeds. The friction between the layers and the turbulence created resulted in the formation of the spot. It is therefore not a fixed structure on the surface, but a dynamic swirling mass of gas. There is a constant flow of gas flowing into the spot and a similar flow leaving the spot. The spot requires the gases flowing in bands around it to remain more or less constant. It could disappear overnight, should the nature of the gasses around it change. Nevertheless the red spot remains stable; its size and shape being kept within the bounds of the attractor for around 350 years. The red spot of Jupiter is also described as a soliton

A community organisation such as a church or sports club can be seen to be working as a strange attractor. Members come and go over the years, sometimes the organisation is more active than others are and new activities may be undertaken. Buildings may be bought and sold, but there is nevertheless an ongoing recognisable identity that remains for as long as the organisation still runs. We can not predict exactly what it will be like at any given time, but there are limits to the organisation and how it is run, beyond which we would say it no longer exists as that organisation. There is no guarantee that the organisation will continue to survive, but as long as it maintains the flow of resources and no unforeseen circumstances arise, it is likely to continue to exist.

Yet another example is a water wheel. If the volume of water filling up the bucket is very slow, the wheel will not be able to overcome friction and remains as a point attractor not moving at all. If the flow speeds up, the wheel turns slowly and predictably, but when the flow increases so that gravity then makes the buckets swing right around then the wheel will stop flowing in an orderly fashion and reverse directions back and forth and in a seemingly randomly fashion, but can be shown to fit the pattern of a chaotic attractor.