 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'll see these distinct, reoccurring, clustering patterns to the distribution of the different cultures. We would also see this clustering within the distribution of political opinions across the different regions of some nation, or again, we'd see this clustering within the distribution of traditional dialects within some region. None of these forms of organization have been planned by some centralized 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 and less perturbed. For any system, we can create what is called a state space that represents all the possible states that that 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 within the phase space. In order to build this state space model, we have to define one or more parameters to the system that we're interested in, where a parameter is simply a measurement of something about that system. So if we were interested in a sales person's finances, we could define a parameter that would measure their income, but this would not be very interesting, it would simply go up and down depending on their sales. So what we're 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'll 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'll see after doing this for a few weeks is some kind of typical behavior. During the week, our sales person is earning some amount of money, then it goes up on a Saturday with lots of sales, but then down on a Sunday when they're not working, and then starts again next week. What we'll typically see is that these different states do not go around every single state in the whole space, but instead they are typically confined to a limited subset of all the possible states. So we can say that this subset of the phase space to that dynamical system that corresponds to its typical behavior is what we call the attractor. A bowl containing a ball may be used to illustrate this concept. If we drop the ball into the bowl, it will move around until it comes to rest at the lowest point. We can say that it is attracted to this point, so each part of the bowl can be regarded as leading to this stationary point, and the whole bowl is what we call the systems basin of attraction. Systems like this bowl 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 that they need some source of energy to maintain their dynamical state. They are continuously inputting new energy and then dissipating it as they cycle through these different states, and they cycle through this process always having to come back to the source of energy that is maintaining their dynamical state, and it is in that cycling that we get all of the different states within the attractor. It's like eating food, doing exercise to dissipate this, and then coming back to eat again. Within social systems, we can think about 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 the counterbalancing forces that are on the system, within its basin of attraction, it can be said to be in a stable equilibrium. That attractor represents an equilibrium for the system. For example, an attractor may represent a social institution of some kind. As we've previously discussed, social institutions serve some function for individuals or society. They are essentially patterns of behavior 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 a pre-existing company is typically easier than creating one's own. Finding 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 stable state. The word bifurcation means splitting or cutting into 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. Mathematicians have borrowed the term bifurcation to describe how a system branches off into new qualitatively different long-term states of behavior. What we're 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 can 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 the Third Estate, after being locked out of 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 monarch, 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 different 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 the other, and this is called Symmetry Breaking. Symmetry Breaking is a phenomena in which critical points decide a system's future state by determining which branch of a bifurcation it takes. 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're 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 with you having to choose to side with the monarch or with the new parliament. Once you've 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 upon the homogeneous belief system of the Catholic Church. With the rise of 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 multicultural societies, with many different religions, philosophies and belief systems. Any individual growing up within this society is no longer held within a single basin of attraction, they are now free to choose from a number of different attractors. 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 logistics map, which is a type of iterative function, meaning we take the output at each iteration and feed it back in to calculate the next value, such as would be the case with population growth, where we take the value to the previous population and feed it into the iterative function to calculate the current population, and then again we'll feed this into the next iteration and so on. We'll 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 that after we have this initial bifurcation we then get more bifurcations happening faster, as they double in rate each period, and this is called the onset of chaos, as we're moving towards a state of more and more attractors, greater and greater differentiation. And this is one way of understanding complex systems. On the left hand side of this graphic of the logistics 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, we get the period doubling with more and more attractors emerging, and this is the chaotic regime of non-equilibrium complex systems that have multiple basins of attraction and can flip between them. This is also 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, and this is chaos. 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 labourers working the land. Culturally we all believed in the same belief system that guided and controlled almost all of our social institutions. Through the process of modernisation, both our culture, society and economy has become increasingly specialised and differentiated. Culturally we've developed a vastly more complex body of knowledge for interpreting our world. Our social institutions have become decoupled from the church, gaining autonomy, and of course economically we've become highly specialised and differentiated within our skills and occupations. This social system that started out relatively homogeneous has gone through many bifurcations and symmetry breaking to become a heterogeneous, complex system with many different attractors. In this module we've been talking about attractors and the fundamental role they play within social dynamics, both with respect to self-organisation and chaos. We firstly gave an outline to the model of a state space that allows us to quickly identify reoccurring patterns within a system's long-term state of behaviour. We talked about how these dissipated systems typically only occupy a small subset of the overall state space as they cycle through some set of states relating to an underlining process of energy consumption and dissipation, and is this subset of states that we called the attractor, that may be interpreted as an agent's path of least resistance. We went on to discuss bifurcations as topological transformations that result in the emergence of two different attractors, requiring agents to adopt a specific state within either attractor resulting in the process of symmetry breaking. Lastly, we saw how this process of this continuous doubling in bifurcations is a universal feature of systems as they move into a chaotic and complex regime resulting in them coming to have multiple attractors in equilibrium.