Brief Introduction to Dynamic Systems Theory

From Neither Brain nor Ghost
By Teed Rockwell
Published 2005 MIT Press

A Brief Introduction to DST

A dynamic system is created when conflicting forces of various kinds interact, then resolve into some kind of partly stable, partly unstable, equilibrium. The relationships between these forces and substances create a range of possible states that the system can be in. This set of possibilities is called the state space of the system. The dimensions of the state space are the variables of the system. Every newspaper contains graphs which plot the relationship between two variables, such as inflation and unemployment, or wages and price increases, or crop yield and rainfall etc. A graph of this sort is a representation of a set of points in a two-dimensional space. Newspapers and journals will also sometimes contain graphs which add a third variable, and thus represents a three-dimensional space, using the tricks of perspective drawing. The state space of the sort of dynamic system studied by cognitive scientists will have many more dimensions than this, each of which measures variations in a different biologically and/or cognitively relevant variable: Air pressure, temperature, concentration of a certain chemical, even (surprise!) a position in physical space. But the mathematics is the same regardless of how many variables the space contains, or the physical or biological process that each dimension is tracking.

However, although these variables define the range of possibilities for the system, only a few of these possibilities actually occur. To study a dynamic system is to look for mathematically describable patterns in the way the values of the variables change and fluctuate within the borders of its state space. The patterns that a system tends to settle into are called attractors, basins of attraction, or invariant sets. In Port and Van Gelder 1995, an invariant set is defined as "a subset of the state space that contains the whole orbit of each of its points. Often one restricts attention to a given invariant set, such as an attractor, and considers that to be a dynamical system in its own right." (p.574) In other words, an invariant set is not just any set of points within the state space of the system. When several interrelated variables fluctuate in a predictable and law-like way, the point that describes the relationship between those variables travels through state space in a path which is called an orbit. The set of points which contains that orbit is called an invariant set because the variations in that part of the system repeat themselves within a permanent set of boundaries.

Port and Van Gelder define "attractor" as " the regions of the state space of a dynamical system toward which trajectories tend as time passes. As long as the parameters are unchanged, if the system passes close enough to the attractor, then it will never leave that region." (p.573). The simplest example of an attractor is an attractor point, such as the lowest point in the middle of a pendulum swing. The flow of this simple dynamic system is continually drawn to this central attractor point, and after a time period determined by a variety of factors (the force of the push, the length of the string, the friction of the air etc.) eventually settles there. A slightly more complex system would settle into not just an attractor point but an attractor basin. i.e. a set of points that describes a region of that space. The reason that these attractors are called basins of attraction is because the system "settles" into one of these patterns as its parameters shift, not unlike the way a rolling ball will settle into a basin on a shifting irregular surface. A soap bubble is the result of a single fairly stable attractor basin, caused by the interaction of the surface tension of the soap molecules with the pressure of the air on its inside and outside. Because a spherical shape has the smallest surface area for a given volume, uniform pressure on all sides makes the bubble spherical. But when the air pressure around the soap bubble changes, e.g. when the wind blows, the shape of the bubble also changes. The bubble then becomes a simple easily visible dynamic system of a sort, marking out a region in space that changes as the tensions that define its boundaries change. To see how these same principles can eventually reach a level of complexity that makes them a plausible embodiment of thought and consciousness, imagine the following developments.

1) The soap bubble could get caught up in an air current that flows regularly so that, even though the soap bubble is not staying the same shape, it changes shape in a repeating pattern. As I mentioned earlier, this pattern is often called an orbit, because the trajectory that describes this repeating change forms something like a loop traveling through the state space of the system. Systems that settle into orbits are usually more complicated than those which settle only into attractor basins which are temporally static, particularly when those orbits follow patterns that are more complicated than mere loops.

2) Instead of having the soap bubble fluctuate in three dimensional space, imagine that it is fluctuating in a multi-dimensional computational state space. As I mentioned earlier, state space is not limited to the three dimensions of physical space, for it can have a separate dimension for every changeable parameter in the system. The most popular example in cognitive science of a system that operates within a multi-dimensional state space is a connectionist neural network. Connectionist nets consist of arrays of neurons, and each neuron in a given array has a different input or output voltage. Each of those voltages is seen as a point along a dimension of a Cartesian coordinate system, so that an array of ten neurons, for example, would describe a ten-dimensional space. But in other kinds of dynamic systems analyses, any variable parameter can be a dimension in a Cartesian computational space. Our friend the soap bubble can be interpreted as a visual representation of the air pressure coming from every possible angle within the three dimensions in physical space, if all other background conditions remain stable. And when the various interacting forces and variables in a dynamic system are designated as dimensions in a multi-dimensional space, it becomes possible to predict and describe the relationships between different attractor basins in that system. This is the most relevant disanalogy between a soap bubble and the more complicated dynamic systems studied by cognitive scientists. Because:

3) A soap bubble has really only one stable attractor basin. Although the attractor space that produces a soap bubble is fairly flexible, the bubble pops and dissolves if too much pressure is put on it from any one side. But in certain systems, there are fluctuations of the variables which can cause the system to settle into a completely different attractor space. These systems thus consist of several different basins of attraction, which are connected to each other by means of what are called bifurcations. This makes it possible for the system to change from one attractor basin to another by varying one parameter or group of parameters, and thus initiate a different complex pattern of behavior in response to that change.

This propensity to bifurcate between different attractor basins is what differentiates relatively stable systems (like soap bubbles) from unstable systems (like living organisms or ecosystems). In this sense, all living systems are unstable, because they don’t settle into an equilibrium state that isolates them from their surroundings. Organisms are constantly taking in food, breathing in air, and excreting waste products back into the environment they are interacting with. We usually think of unstable processes as formless and incomprehensible, but this is often not the case. Certain unstable systems have a tendency to settle into patterns which still fluctuate, but fluctuate within parameters that are comprehensible enough to produce an illusion of concreteness. When the various forces that constitute the processes shift in interactive tension with each other, a basin of attraction destabilizes in a way that makes the system bifurcate i.e. shift to another basin of attraction. This kind of system is sometimes called multi-stable, because its changes between various basins of attraction are predictable and (to some degree) comprehensible. A complete pictorial graph of a dynamic system of this sort would resemble figure 4. It would show interlocking computational spaces whose transitions were governed by the relationships between the constituting forces of the system.

This process of bifurcation bears a significant resemblance to the switching between possible branches of decision trees which is the fundamental cognitive process performed by computer languages. And this kind of decision-making is an essential part of many extra-cranial aspects of skillful sensory motor activity, a fact which plays havoc with Descartes' distinction between the so-called automatic functions of the body and the rational decision making of the mind.