Bifurcations & Attractors
An attractor is a set of states towards which a system will naturally gravitate and remain cycling through unless perturbed.1 For any given system we can create a state space representing all the possible states that the system might take, the attractor is then a subset of those states that correspond to the system’s typical behavior. A bifurcation is a qualitative topological transformation in this state space resulting in a spitting of this attractor into two distinct stable attractors.2
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. 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. 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. This subset of the phase space of the dynamical system corresponding to its typical behavior is the attractor.
Basin of Attraction
A bowl containing a ball may be used to illustrate the concept of a basin of attraction. 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, i.e 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 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.
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 behaviours and some equilibrium state.
The word “bifurcation” means splitting or cutting in two. If a river divides into two smaller streams, that is a bifurcation. If you split a company into two divisions, that is 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 behaviour. What we are interested in complex systems is primarily a bifurcation within an attractor, meaning instead of having just one attractor in the state space, a bifurcation will now give us two attractors and that means two stable sets of states that the system may cycle through. We could cite 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 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.