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. Our heartbeat is regulated by an attractor so that there are patterns of heartbeats that are possible and others that are not because they would not sustain life.
An attractor has limit values, like limits on how fast or slow a heart may beat, which define the boundary within which the values must land to be in the attractor, although the boundary marks are typically 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, or how we lived our life, we die at the end of our life. Water in a bathtub has the plughole as a point attractor. There is only one place for the water to go.
The phase space is all the possible states of the system mapped so it looks like territory with higher places space that are more complex, but harder to reach or lower places, where things tend to naturally go. A point attractor on the phase space will be a basin. If a small ball is placed anywhere in 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, Places outside the cycle are not possible. If a system is forced away from an attractor, it will be drawn back into it.
An example of a limit cycle is a predator-prey system. In Canada, the lynx eats hares. When there are plenty of hares about, the number of lynx grows because they have so much food. Eventually, the lynxes have eaten so many hares there are not so many left, and the number of lynxes drops. But when the number of lynxes drops, the number of hares increases again because they are not being eaten. This creates a cycle where the populations of hares and lynxes change in a regular pattern. While we cannot pick a time at random and predict the population of lynxes and hares, we can identify the limit cycle and predict how the populations will change over time. We also know that we cannot get high populations of lynxes and hares at the same time. The lynxes would eat the hares until it is back in the cycle.
The seasons form a limit cycle. We cannot predict the exact temperature of any day, but if we know the season, we can know that certain states are more likely than others.
On the fitness landscape, a limit cycle would be seen as a valley that goes around 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 to some point in the round track at 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 pendulum added on the end of the other forms a torus-shaped cycle when charted.
The fourth type of attractor is called a strange attractor or a chaotic attractor. A strange attractor never repeats itself (or it would be a 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 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 comprises 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 surrounding gasses change. The red spot of Jupiter is also described as a soliton.
A community organisation such as a church or sports club can operate as a strange attractor. Members come and go over the years. Sometimes the organisation is more active than others 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 cannot 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 that fits the pattern of a chaotic attractor.