THAT’S NO MOON: ATTRACTORS, PATH DEPENDENCE, ENTRAINMENT

EPISODE 28 MUSIC IN PHASE SPACE

Obi-Wan Kenobi: That’s no moon… It’s a space station!
Han Solo: It’s too big to be a space station!
Luke Skywalker: I have a very bad feeling about this.

— Star Wars Episode IV: A New Hope

What is an attractor?

An attractor is a set of states towards which a system will naturally gravitate and remain cycling through unless perturbed. 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.

Imagine a beach. At one end is a pier and the other is a rocky point. Two ice cream vendors arrive to sell their wares and decide to locate themselves so they are equidistant from the pier, the point and one another. By pure chance, vendor A gets the first group of customers. So as not to miss out on business, vendor B moves a bit closer to vendor A. Now vendor B has customers, so vendor A decides to move closer to vendor B. Over time they creep toward each other until they are both side by side. The resulting cluster is called the attractor. They are not attracted to a particular grain of sand in the middle of the beach. Rather, their interaction results in the attractor pattern forming.

Attractors are portions or subsets of the phase space of a dynamical system, invariant under the dynamics, towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct basins of attraction.

Social Attractors & Chaos Theory

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 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 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.

Cultural Attractors

In 1967, Richard Dawkins introduced the idea of a meme: a unit of cultural transmission capable of replicating itself and of undergoing Darwinian selection. The case of the meme idea illustrates a general puzzle. Cultures do contain items — ideas, norms, tales, recipes, dances, rituals, tools, practices, and so on — that are produced again and again.

Rounded numbers are cultural attractors: they are easier to remember and provide better symbols for magnitudes. So, we celebrate twentieth wedding anniversaries, hundredth issue of journals, millionth copy sold of a record, and so on. This, in turn, creates a special cultural attractor for prices, just below rounded numbers — $9.99 or $9,990 are likely price tags — , so as to avoid the evocation of a higher magnitude.

PATH DEPENDENCE

Path dependence refers to the fact that often, something that seems normal or inevitable today began with a choice that made sense at a particular time in the past, but survived despite the eclipse of the justification for that choice, because once established, external factors discouraged going into reverse to try other alternatives.

The paradigm example is the seemingly illogical arrangement of letters on typewriter keyboards. Why not just have the letters in alphabetical order, or arrange them so that the most frequently occurring ones are under the strongest fingers? In fact, the first typewriter tended to jam when typed on too quickly, so its inventor deliberately concocted an arrangement that put A under the ungainly little finger. In addition, the first row was provided with all of the letters in the word typewriter so that salesmen, new to typing, could wangle typing the word using just one row.

Quickly, however, mechanical improvements made faster typing possible, and new keyboards placing letters according to frequency were presented. But it was too late: there was no going back. By the 1890s typists across America were used to QWERTY keyboards, having learned to zip away on new versions of them that did not stick so easily, and retraining them would have been expensive and, ultimately, unnecessary. So QWERTY was passed down the generations, and even today we use the queer QWERTY configuration on computer keyboards where jamming is a mechanical impossibility.

Bifurcation

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. This bifurcation and symmetry breaking process is pervasive across many different types of systems, 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.

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.

To help us understand what this might mean let’s think about Napster as an example, Napster was envisioned as an independent peer-to-peer file sharing service by Shawn Fanning. The service operated between June 1999 and July 2001. Its technology allowed people to easily share their MP3 files with other participants. Although the original service was shut down by court order, the Napster brand survived after the company’s assets were liquidated and purchased by other companies through bankruptcy proceedings.

History

Although there were already networks that facilitated the distribution of files across the Internet, such as IRC, Hotline, and Usenet, Napster specialized in MP3 files of music and a user-friendly interface. At its peak the Napster service had about 80 million registered users. Napster made it relatively easy for music enthusiasts to download copies of songs that were otherwise difficult to obtain, such as older songs, unreleased recordings, studio recordings, and songs from concert bootleg recordings.

High-speed networks in college dormitories became overloaded, with as much as 61% of external network traffic consisting of MP3 file transfers. Many colleges blocked its use for this reason, even before concerns about liability for facilitating copyright violations on campus. Prior to this event we had a single attractor within the phase space to the music industry, it was an all activity was beneath and in relation to the Big Four, Napster 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 regime of the record industry, 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 Record Industry or Napster, 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.

Types of attractors

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Point 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.

such as the sloshing water in a glass, or the bottom center of a bowl contain a rolling marble. Physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the nonlinear dynamics of stiction, friction, surface roughness, deformation (both elastic and plasticity), and even quantum mechanics. In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly hemispherical, and the marble’s spherical shape, are both much more complex surfaces when examined under a microscope, and their shapes change or deform during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains. There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are considered stationary or fixed points, some of which are categorized as attractors.

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.

Limit cycle

A limit cycle is a periodic orbit of a continuous dynamical system that is isolated. Examples include the swings of a pendulum clock, and the heartbeat while resting. Example, a planet orbiting around a star,

Under this Attractor you cycle back and forth from a set of two or more activities. Although not as simple and direct as the Point attractor, there is still regularity and simplicity to the cyclic events. An example is a desire to sleep at the end of a day, which when gratified naturally leads to a desire for activity at the beginning of a new day, followed much later by a desire to sleep again, etc. In Nature it can be seen in many ways; for instance, the predator prey systems where the respective predator prey populations cycle up and down in relation to the other.

This finding is very relevant to social systems, and we can see this dynamic everywhere we look, most clearly perhaps in politics where oscillations between ‘Right’ and ‘Left’ occur at regular intervals, the policies of the party in power descending into chaos, whilst the opposition position estabilises and gains control of the ordering process following an election.

The net result is that the system self-organizes to attain a dynamical balance, which we call ‘edge-of-chaos’, where different areas of the system exist in different states,

TORUS ATTRACTOR

For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies

A system which changes in detailed characteristics over time but does not change its form will have a trajectory which will produce a path looking like the doughnut shape of a torus. Example, picture walking on a large doughnut, going over, under and around its outside surface area, circling, but never repeating exactly the same path you went before. The torus attractor depicts processes that stay in a confined area but wander from place to place in that area.

An example of the Torus attractor at work would be a more complex set of attracting events which occur to a person on many levels over a course of a year, and repeat again, year in and year out. For example, a desire to golf each summer, hike each fall, and eat and drink too much on holidays. In Nature it is shown for instance by the complex interaction of a number of interdependent species: the population of one predator species relates to that of the prey of its prey. For example, the size of the insect population effects the size of the frog population, which effects the size of one of their predators, the trout, which in turn effects their predators, the pike.

Strange Attractor

An attractor is called strange if it has a fractal structure.This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist.

An attractor in phase space, where the points never repeat themselves, and orbits never intersect, but they stay within the same region of phase space. The Strange Attractor can take an infinite number of different forms.

Social media could be considered a current dominant manifestation of strange attractors. One that has become familiar to us with our everyday usage of it and yet it continues to change daily. With changes in structure, content and the users involved.

In real life when we need to look to availability, emotions, response to design, likes and dislikes, morals and behavioral norms, inspirations, aspirations, expectations and rewards etc. The effect of influencers should be of particular of note.

A significant area of interest for business is in trends, hits, fads and business cycles.

ENTRAINMENT

One day, Huygens noticed a set of pendulum clocks placed against a wall happened to be swinging in perfect synchronised chorus. Nothing in mathematical description could explain this propagation of order from one pendulum to other; clocks could not be that accurate. Huygens surmised the clocks were coordinated by vibrations transmitted through the wood. This phenomenon in which one regular cycle locks into another is now called entrainment, or mode locking. Entrainment also explains why stock market components are like flocking birds, shoaling behavior of fish, the swarming behavior of insects and herd behavior of land animals. why the Dow 30 performance is connected to Sensex 30, gold is connected to dollar and a host of other flocking performances in the capital market.

Though there is a pattern in the flocking of birds in the sky, the form is not always predictable. But there is a difference between random and unpredictable. The weather is unpredictable but not random. Unpredictable events or systems can be described as those we are unable to forecast or only able to partially forecast, due to a lack of information.

Another universality seen across data assets is extreme reversion. Whenever the bird separates from the flock, it is strangely attracted back into the fold. The outliers don’t remain outliers; the attractor keeps them together. Call it miracle but the worst outliers invariably deliver.

Attempts to reduce systems to one behavior overall are doomed to failure, we must accept that ‘sets’ of diverse dynamics always simultaneously exist, reminiscent of the diversity of features and attractor types present in the structure of the Mandelbrot Set.

Non-equilibrium systems, like society, are not comprised of single formulae, they are fractal and have diverse structure on many scales. They are composed of many autonomous elements, each operating with many different values. In the dynamics of these situations we get strange attractors, not point or cyclic ones — a society existing in such a limited attractor would be a ‘dead’ or ‘dying’ society ! In real societies we cannot predict ‘exactly’, solutions are nonlinear, there is a sensitivity to initial conditions, i.e. to history. There exists heterogeneity, multiple interacting dynamics, these systems are often non-deterministic — dependent upon ‘random’ events. For such systems a new type of science is required, needing a new set of valuation techniques, a metascience of interconnected reality and values, which we pursue further here.

When we look at social situations we must be quite clear about the differences between these and those simpler systems typically studied in the physical sciences. In the latter it is usual to concentrate upon equilibrium solutions and to seek out point or cyclic attractors — labelling any situation incapable of being stated in such linear and reductionist terms as ‘unscientific’ and ignoring it. In such ‘science’ data points found outside the ‘box’ of predetermined expectations are actually discarded, and treated as ‘experimental error’ — making this a self-fulfilling and closed worldview. Physical sciences, typically, look here for one ‘formula’ at a time, one isolated problem at a time. Most work in the current psychological and sociological sciences tries to ‘ape’ this sort of technique, either discarding all the ‘complexity’ to look at a narrow aspect, or using statistical techniques to look at the undifferented whole — in both cases compressing the system to a single parameter (and arguing endlessly between themselves about which one to use !).