Attractor States

What's so Attractive?

Often a system has the capacity to unfold in many different ways - it has a large 'possibility space'. That being said, there can be regions of this possibility space that exert more force or 'attraction' for the system.

In some kinds of systems these zones of attraction exist because of pre-determined energy minimizing characteristics of these regimes. For example, if we blow a soap bubble it 'wants' to become a sphere: this is the state where there is the most volume for the least surface area, and therefore also the best configuration for soapy molecules given that it locates them in their lowest energy state: one that best balances competing forces, being the expansion forces of the air pushing the system outwards, and the resistance forces of the soapy solution not wanting to waste any surface area. The spherical shape of the bubble is thus a kind of pre-given, and when we blow a bubble this it is this shape - rather than a cube or a conical form, that we can safely anticipate the form will take. 

Similarly, if we toss a marble in a vortex sphere at a science museum we know it will spin around the surface, but then ultimately make its way down to the bottom: this is the state of minimum resistance to the forces of gravity acting upon it.

It is this 'minimizing behavior' that is characteristic of attractor states - of all possible states within a given system Phase Space (the space of all possibilities) some regions may require less energy expenditure to move towards others. We will see that there can also be systems that have more than one such minimizing regime.

Lock In!

While the two physical systems described above have natural attractors, there are also social system dynamics that can cause similar attractor dynamics to arise.

In these scenarios, attractor states are not necessarily pre-determined by natural forces, but can instead emerge over time, as the system evolves, and in light of Feedback forces.  That said, once present they can reinforce themselves by constraining subsequent actions of agents forming the system.

Example:

We can think of Silicon Valley as being an emergent attractor for tech firms, that has, over time, reinforced its position. What is interesting about this example, is that even though it comes from the social sciences rather than physical sciences, in some way the same minimizing principle applies - it is just a different form of minimization that does not have to do with the laws of nature, but instead the social laws of human interaction.

To put this another way, once Silicon Valley established itself as the main tech hub, any new entrants to the tech field could, in principle, have chosen to locate themselves elsewhere - there were multiple locational possibilities win Phase Space. However, if they were to choose these other locations, they would be far more likely to encounter additional "resistance" or frictions that would inhibit success. This is because these non-silicon valley sites would lack factors such as of supporting infrastructure, abundant knowledge spillovers, experienced and readily available workers, etc. In a sense, the smoothest, least resistant course of business action for a technology firm is thus to locate where these kinds of external inputs are most easily accessed: a 'state of least resistance' - which in this case equates to Silicon Valley.

The emergence of such clusters of expertise is not limited to Silicon Valley. We often see that groupings of similar business co-locate in space (referred to as agglomerations), rather than distributing themselves evenly across a region. In a particular city we will see groupings ranging from jewelry stores, or cell phone service providers, or bridal salons, tending to coalesce in co-located groupings.

The precise locations of these groupings is not something that is established in advance in the way that the spherical shape of the soap bubble is. Instead, in these instances it is the processes of Feedback that, over time, reinforce minor locational advantages, such that the kind of spill-over advantages discussed in the Silicon Valley example lead businesses that co-locate to have a better chance of success compared to their far-flung competitors. 

Once these kinds of concentrations of expertise have coalesced in a particular region, it then attracts new entrants to the field, in the same way that the spherical form attracts the soap molecules. Any systems that enter into this kind of regime, where new behavior is directed according to what has occurred before in ways that are constraining and directing, can be considered to have entered into "Enslaved States" (a term popularized by Hermann Haken. The concept of 'Enslavement' captures the notion that certain attractor states can emerge from agent interaction, and once present,  will constrain the future action of these agents and all that come after them. The same idea is referred to as 'Lock-in' in the field of Evolutionary Economic Geography.

Shake it Up

We can see that in the example of the soap bubble, and the example of Silicon Valley, we have two very different kinds of system that are nonetheless both still trying to limit unnecessary energy expenditures. For the soap, the concern is minimizing surface tensions or stress, for the business owner, minimizing the tensions and stresses involved with finding good employees, or  access to good internet, etc. In this way the dynamics, while at first completely different, nonetheless run parallel. What is different is that in the human system the 'laws' at play are not stable over time. What might be best practice at one instant is not necessarily best practice at a later time. This is the risk of Lock-in:  that systems begin to perpetuate themselves beyond the point that they were helpful (the QWERTY key board, designed to slow down typists to ensure that the mechanical typing hammers would not jam, is a great example of this kind of lock-in).

In these kinds of lock-in systems not governed by physical laws, it is occasionally worth 'shaking the system up' in order to see if it can be dislodged from a weak regime and encouraged to explore alternative behaviors. This is described as introducing a system Perturbation, a disturbance intended to jostle a system and then see what it settles back into.

Example: 

For much of human history, the most effective way for individuals to access goods was for them to converge towards a central market-place. This was the area for trade, and by being centralized and co-located, efforts to find goods could be minimized on the part of the consumer, and efforts to find customers could be minimized on the part of the seller.  This was the most "fit" way of achieving the goal of the acquiring and dispersing goods.

In recent decades, this model has begun to shift on its head. With the advent of information technologies, combined with innovations in transport logistics, it has become increasingly viable for companies to deliver goods directly to the homes of consumers. Rather than coming to a central market-place, goods are able to move directly from manufacturer to consumer. Frictions about what is needed where have been reduced, and costs and energy associated with physical markets vs virtual markets have been similarly reduced. 

We can think of each of these regimes of behavior and as each two separate attractor basins within a variegated possibility space of goods acquisition and dispersal strategies.  With changes in technology, one basin of attraction has, over time become more viable (therefore deeper), and the other seems to have shrunk back in relevance and depth. We seem to have arrived at a tipping point today, where the minimizing forces favoring e-commerce vs physical commerce have shifted. That said, the legacy system tends to persist (old habits die hard).

Enter a global pandemic: this is a great example of a system perturbation, which shakes up standard patterns of behavior. Indeed, Covid caused many people who had never shopped online to try this behavior, and realize that it does, indeed, minimize effort in new ways. This kind of system disturbance has moved many people out of their taken for granted regime of behavior, and caused them to move into new regimes. 

We can see from this example how a system perturbation can act as a kind of productive 'shock' that, if large enough, is able to move a system out of a prior attractor state and potentially into a new regime.

Multiple Attractors

In discussing the example above, we slipped in the idea that a system may have more than one 'well' or basin of attraction. It is worth exploring this a bit more, since we can imagine different kinds of possibility spaces - some that only have one deep well to which everything will ultimately  tumble (a single attractor like that which the pendulum moves towards), others can have multiple attractors, some deeper, some shallower, with a system able to explore multiple regimes of behaviors within the space.

Further, complex systems can sometimes oscillate between attractor states, both of which are equally viable. This can be described as a system having Multiple Equilibria. The example of Benard Rolls is a case in point - liquid is heated from below, and forces churn the water molecules so as to cause them to minimize resistance by moving into a "roll" pattern. That said, the direction of the roll -cascading left or cascading right- or two equally viable minimizing behaviors, both of which the liquid can move into. The system therefore has multiple equilibria

In addition, we can have systems that oscillate between attractors, rather than settling into a specific regime. An example would be a predator/prey system, where the population numbers of each species each rise and crash in recurring patterns over multiple generations. In this case, two attractors are coupled whereby as one intensifies (the prey reproduces a lot), in generate a response in another part of the system that is counterbalancing  (the predator finds a ready food source and is able to reproduce a lot). This creates a back and forth oscillation between high prey and high predator numbers, with each regime counterbalancing the other.

The same dynamics can be seen in what are known as chemical oscillators, where we have a phenomena of multiple attractors described as follows:

• a reaction intensifies certain chemical behaviors;
• beyond a certain threshold  these behaviors catalyze a new, counter behavior;
• this counter behavior intensifies...;
• beyond a certain threshold this counter behavior catalyzes the first behavior;
• etc. 

The result of these reactions can be quite surprising, as seen below!

Check out the Multiple Attractors in the Briggs Rauscher chemical oscillator.

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