Social attractors define a specific subset of states that a social system may take, which corresponds to its normal behavior towards which it will naturally gravitate. When we look at many different types of social systems we see distinct patterns of clustering, distinct substructures that have synchronized their states, if we look for example at the distribution of ethnic groups across many multicultural cities we will see these distinct recurring clustering patterns of the different cultures. We would also see this clustering within the distribution of political opinions across the different regions of some country, or again the clustering of traditional dialects. None of these forms of organization have been planned by a central authority they are all examples of emergent phenomena. All of these different clustering patterns are examples of attractors which are central to understanding the process of pattern formation within nonlinear systems.
An attractor is a set of states towards which a system will naturally gravitate and remain cycling through unless perturbed. For any system we can create what is called a state space, that is all the possible states that the system might take. A state space also called a phase space is a mathematical model in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.
In order to build this state space model we have to define one or more parameters to the system that we are interested in, where a parameter is simply a measurement of something about the system. so if we were interested in a sales person’s finances we could define a parameter, to measure their income, but this would not be very interesting it would simply go up and down depending on their sales, so what we are typically interested in then is the relationship between two or more different parameters. So we might define another parameter to their overall savings or wealth. Now at each day we will take a sample of both of these parameters, putting a dot at the corresponding value and stay doing this over a period of time.
What we will see after doing this for a few weeks is some kind of typical behavior, during the week they are earning some amount, then it goes up on Saturday with lots of sales but then down on Sunday when they are not working and then starts again the next week. What we will typically see is that these different states do not go around every single state in the whole space but are confined to a limited subset of all the possible states. So we can say that this subset of the phase space of the dynamical system corresponding to its typical behavior is the attractor.
A bowl containing a ball may be used to illustrate the concept. The ball will move around the bowl until eventually, it comes to rest at the lowest point. We can say that it is ‘attracted’ to that point, so each part of the bowl can be regarded as leading to that stationary point, and the whole bowl is what we call the system’s basin of attraction.
Systems, like this ball, are typically held within their attractor because of the different forces placed upon them by their environment, an animal stays on a particular patch of fertile land and does not stray too far from it because it needs to eat, a person gets up and goes to work every day because they need the money to support themselves. What is happening here is that these dynamical systems are dissipative, meaning they need some source of energy to maintain their dynamic state, they are continuously inputting new energy and then dissipating it, and they cycle through this process always having to come back to the source of energy that is maintaining their dynamic state, and it is in that cycling that we get all the different states within the attractor.
Within social systems we can think of attractors as representing the course of least resistance for a person or social group at any given time, they remain within their current configuration because of inertia. Due to these counterbalancing forces that are on the system within its basin of attraction, it can be said to be in a state of equilibrium.
For example an attractor may represent a social institution of some kind, social institutions serve some function for individuals and society, they are essentially patterns of behaviour or belief that exist within a given society in order to serve basic human functions, institutions represent pre-existing solutions to given social challenges both personal and social, as such they are the course of least resistance for individuals within that society, working for an existing company is typically easier than creating one’s own, adopting the values of one’s society is typically much easier than reading a big pile of philosophy books to figure out one’s own beliefs and values. These attractors then keep social actors within a well-defined set of behaviors and some equilibrium state.
The word “bifurcation” means splitting or cutting in two. If a river divides into two smaller streams, that’s a bifurcation. If you split a company into two divisions, that’s a bifurcation too. Mathematicians have borrowed the term bifurcation to describe how a system branches off into a new qualitatively different long-term state of behavior. What we are interested in here is really a bifurcation in these attractors, so instead of having just one attractor in our state space, a bifurcation will now give us two attractors and that means two stable sets of states that the system may cycle through.
To help us understand what this might mean let’s think about the French Revolution as an example, in particular what is called the tennis court oath which was a pivotal event during the first days of the French Revolution, when Third Estate, after being locked out from the government, made a makeshift conference room inside a nearby tennis court, calling themselves the National Assembly they went on to form the new political republic of France. Prior to this event we had a single attractor within the political state space to the nation, it was an absolute monarchy all political activity was beneath and in relation to the monarch, this tennis court oath was then a bifurcation in the topology as a new attractor formed. Any agent within this state space after the bifurcation is going to have to choose one of the attractors, whereas previously before this bifurcation everyone was under the same political regime of the monarch, that is to say everyone had a symmetric homogeneous state, but now that we have two attractors people have to choose one state or another and this is called symmetry breaking.
Symmetry breaking is a phenomenon in which critical points decide a system’s fate, by determining which branch of a bifurcation is taken, such transitions usually bring the system from a symmetric but disorderly state into one or more definite states, as such symmetry breaking plays a major role in pattern formation as we are now getting differentiation and some form of organization, that is to say, that there is now some relationship between these different parts. To continue on with our previous example, this symmetry breaking would correspond to you having to choose to side with the monarch or the new parliament, once you have made this choice you are now within one of the two basins of attraction, you have differentiated your state with respect to others and out of everyone going through this symmetry breaking we will start to get a new pattern of organization forming.
As another example we might think about the massive cultural revolution that took place within Western society as we moved into the modern era, prior to the scientific revolution and The Enlightenment this society was based on the homogeneous belief system of the Catholic church, with the scientific secular vision of the world we had a bifurcation in this cultural state space and ever since we have had many more bifurcations until today we live in a multicultural societies, with many different religions, philosophies and belief systems, an individual growing up in this society is no longer held within a single basin of attraction, they are free to choose from a number of different attractors.
Onset of Chaos
This bifurcation and symmetry breaking process is pervasive across many different types of systems, this process is most clearly expressed in what is called the logistic map, which is an iterative function, meaning we take the output at each iteration and feed it back into calculate the next value, such as with population growth, where we take the previous population and feed it into the iterative function to calculate the current population and then that again will feed into the next iteration and so on. We will not go into the details of this logistics map but what it tells us is that there is what is called a period doubling in the rate of bifurcation, meaning after we have this initial bifurcation we then get more bifurcations happening faster, doubling in rate each period and this is called the onset of chaos as we are moving towards a state of more and more attractors, great and great differentiation.
This is one way of understanding complex systems, on the left-hand side of this graphic of the logistic map, we have systems with a single equilibrium, which is characteristic of simple linear systems, we then have a bifurcation as we get the emergence of two attractors, from here on we get the period doubling with more and more attractors emerging and this is the chaotic regime of nonequilibrium complex systems that have multiple basins of attraction and can flip between them. And this is one way of understanding what is called chaos, where chaos means sensitivity to initial conditions, two things that started out almost exactly the same, diverge and ultimately end up in totally different basins of attraction. No matter how close together two states were initially and no matter how long their trajectories remain close together, at any time they can suddenly diverge going in completely different directions.
Going back to our previous example about the development of Western society, we might think about how at the beginning of the modern era we were all relatively economically, socially and culturally similar, economically almost all of us were manual laborers working the land, culturally we all believed in the same belief system that guided and controlled all social institutions. Through the process of modernization, both our cultures, society and economies have become increasingly specialized and differentiated, culturally we have developed a vastly more complex body of knowledge for interpreting our world, our social institutions have become decoupled from the church to gain autonomy and of course economically we have become highly specialized and differentiated within our skills and occupations. This social system that started our relatively homogeneous, has gone through many bifurcations, and symmetry breaking to become a heterogeneous complex system with many different attractors.