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Attractors and Mental Processes
by Ray Eales

Do what thou wilt shall be the whole of the Law.

For thousands of years mystics have attempted to catalogue human meditative states. The Hindu Yogis were very active in this work. Their language for subtle differences in and the partitioning of these states of consciousness is unrivaled in scope. For the westerner it does have its drawbacks however. There is a very sizable language barrier due to the difficulty of translating these subtleties of meaning into a western language. Most western languages do not have similar concepts built into them. European languages, including American English, often adopt a westernized form of these terms with definitions supplied by the various translators of the works in which they appear. This is all very well except that in this situation the western student is faced with the task of sorting through multiple and often conflicting translations of these terms into the various western languages. Thus very often the western student is on his or her own when it comes to learning what terms like Dhayana, Pratyhara and Samadhi mean. (Note: even the spelling of these terms varies greatly so a person reading this document may or may not recognize the terms to which I here make reference. If the reader has not studied Legge, Crowley and Motta very likely the spelling of these terms will seem off to that reader.) Well admittedly there are more formidable tasks in the repertoire of the student of the mysteries than a little translation trouble and thousands before now have come to a pretty fair conclusion as to the subtle meanings of these terms. But still there is perhaps a science more familiar to the western student which can itself throw some much needed light on this subject. I'm talking about mathematics. Not just any mathematics but the mathematics of linear and non-linear systems.

There is a concept unique to these mathematics which lends itself very nicely to description of certain states of consciousness and it is called the attractor. Attractors come in several varieties the fixed attractor, the limit cycle attractor and the strange attractor. All three types may well be important in assisting us to understand more clearly our own mental processes.

The idea of the attractor is actualy quite an old idea (relative to "new" mathematics - older than Einstein and youger than Newton). The fixed attractor, such as a pendulum whose movement is mapped in phase space (see fig. 1), is also not new concept but is rather a re-interpretation of the classic pendulum system moving in a vacume. The concept of phase space while a comparatively new idea is a particularly attractive model for the mapping of various kinds of mental processes. (Note: I will go into more detail concerning phase space a little later in this lecture.) First we must decide what we are going to call the brain states which we are going to examine. Although the material on attractors and phase space can offer much to the study of the normal personality of an individual that study is not the sole purpose of this lecture. Nor will I confine myself to speculations on what I will call the Meditative state. This lecture will contain a little from several worlds. The Meditative state of consciousness is defined for this lecture as a state of mentation in which perturbations from outside influences are at a minimum and the mental process is either locked on one particular thought or has annihilated thought altogether. The student may well ask what exactly does "annihilated thought altogether" mean? It is meant to describe a state of "mentation" in which there is some positive potential, a tendency to go forth but no concept of the form of that going. It is a state of superlative motion without position. This is of course a contradiction on several levels but so is this particular state of mind or I suppose I should add in Crowleyesque fashion, not mind.

What exactly is Phase space? A fair question so lets start further back in the basics. Consider a pendulum system in a vacuum for a moment. Now take a look at the system whose horizontal and vertical components represent momentum and position in the drawing labeled Figure 1. Momentum is the speed at which the pendulum swings. As shown in figure 2 the speed at which a pendulum swings varies with its position. A system which shows a variation in two distinct areas or sets of attributes is said to have two dimensions of freedom. Thus the circle in figure 1 above shows this system's 2 dimensional phase space diagram. Remember that we said that this pendulum system was functioning in a vacuum, remember also that in a vacuum there is no friction, thus no decay in momentum. Since there is no decay in momentum there is also no decay in position and therefore the phase space stays perpetually in the same form; a circle. A pendulum system in the real world would be subject to friction and thus as its momentum decayed from the action of friction so too would its position vary. This variance would take the form of an inward spiral. The central point of the spiral would constitute the state of absolute rest or the central steady state attractor.

From this description it will be an easy step to visualize that an helix is a phase space diagram with three degrees of freedom. The classic helix is like a pendulum system in a vacuum which is being moved at a steady rate in a third dimension:

Phase space is then the plot of a systems movement through 1 or more dimensions and an attractor is the point or shape to which that movement tends. Well if phase space is simply a place where one or more dimensions meet and form a pattern then it will be easy to see how this concept can be applied to many processes including mental processes.

We have talked a little about fixed point attractors and about limit cycle attractors. We have also introduced the concept of phase space. Until fairly recently in the world of attractors that was all there was, nothing more. You had order or chaos and a mysterious, almost paranormal, transition between those two states. Well, the bifurcation diagram shown below and Feigenbaum's constant cleared up the transition part of it by showing a certain universality among all smooth flows transiting to chaotic or turbulent flows. Fegeinbaum's work also showed how the points at which a system transited from laminar flow to turbulent flow could be predicted. This prediction of and finally proof of universality in these phenomenon is still, it seems to me, a matter of controversy among scientists. To persons like myself it is only a confirmation of what I intuitively knew had to be the case all along. (But :"metaphysicians" like myself are still quacks, and frauds and so on, we, if we teach have dupes and stooges not pupils. The little dogs of reason bark at us but we keep telling them to go fetch.... sometimes they do and then they come back with an object of great value.....) But physicists like Richard P. Feynman were still left with the feeling ".....that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do?" (Partial quote from Chaos Making of a new Science, chapter on Strange Attractors.) In 1963ev Edward Lorenz was doing some research on turbulence in fluids. He had come up with three equations which modeled convection. Results of these three equations were mapped using the technique of a phase space diagram shown here and when this was done a funny thing happened... this diagram emerged:

There are some brain and cognition researchers who believe that the attractor describes the way the brain works. Ilya Prigogine says something like "the brain must be largely irregular or else you have epilepsy" he refers to the fact that the electrical impulses of the brain during an epileptic seizure form regular wave-like patterns rather than the irregular patterns of normal mental activity. Also some researchers at UCLA (Don Walters and Alan Garfinkle) have done research with mathematical models of the firing patterns of neurons which indicate that once several neurons are linked together that a "low level chaos with implicit order develops". This means that although specific firing patterns are highly unpredictable that these patterns tend to one shape or another. Scientists at Los Alamos Center for Nonlinear Studies have been working out the fractal dimensions and the strange attractors associated with different levels of anesthesia. In "The Turbulent Mirror" Briggs and Peat pose the interesting question "Could the brain's overall expression, the personality, also be a strange attractor?" Well various researchers have been finding attractors on all scales of brain function, from formation of thoughts to the storage patterns of memory to the distribution of language functions in the brain and even as mentioned above the firing patterns of small groups of neurons, so the question it seems is not really very startling.

Let us suppose for the moment that we are all involved in a mind training practice known as Raja Yoga (Union through control of thought). If we were performing this exercise then each day we would sit for a pre determined time span and stare at one of five simple symbols. A red triangle, a yellow square, a blue circle, an indigo oval or a silver crescent. We would place one of these symbols in our field of view and sit in a comfortable posture (an Asana) and fix our gaze on the chosen symbol. We would start some kind of timer to tell us when our pre determined time span was over. We would count the number of times that our mind strayed from the task, each time the object changed color, size, position, shape and so on. Every time we caught the mind wandering from the task we would mark a break. At the end of the allotted time span we would count the number of breaks and record this number in our journal. Slowly over some weeks we would draw the symbol into our mind's eye and hold a purely mental picture of the chosen symbol in our mind's eye rather than staring fixedly at the physical image. After some months of practice we should see three things happen:

1. The number of breaks should decrease.
2. The time span of perfectly controlled attention should increase.
3. We should internalize the symbol, bring it into our mental field of vision rather than the physical field of vision.

Once we had reached some convenient number of breaks Vs time we would go on to a more demanding practice such as viewing some kind of moving system. Thus we would train our minds to hold images in the mind's eye on demand without the tendency for the mind to become bored and wander off to other stimuli. The question would be if we plotted number of breaks Vs time would we find order or chaos? The example I will give is of a practice lasting 90 days done morning and evening of those 90 days. The PM exercises for the first thirty days on five day averages went as follows: 25, 23.8, 26, 18.6, 20.8, 17.2.

Next we see a bar chart of the AM, PM and averaged break totals for each five day period over 90 days.

It seems fairly evident that at first there are wild fluctuations which gradually settle into a basin of attraction which is roughly a period two bifurcation. That means that the number of breaks oscillate between about 13 and 16. Figure 7 shows a typical bifurcation phase diagram, notice how an initial state follows a regular path until a certain point at which time the path takes two different routes. This is followed by another bifurcation and another until a state of indeterminacy is reached. This state of indeterminacy has within it stable zones but mostly there are only vague tendencies to peruse a path from the final bifurcation set. These tendencies are shown by the more solid lines within the chaotic regions.

This bifurcation diagram is only one of the many ways to observe a system. If enough iterations of the exercise could be plotted then it would be possible to plot an attractor as well. My bet is that this attractor would be a strange attractor. For the purpose of this lecture I am going to call the meditative state a dynamic system and I am going to say that the mind's activity resembles states of flow. A few notes on dynamic systems will be in order at this point in the lecture. Remember that we said that in the classical study of dynamic systems a system is said to have one dimension for every degree of freedom exhibited by the system. Thus the pendulum system shown above had 2 degrees of freedom and its phase space diagram was shown to be a circle. When we added a third dimension to the system the phase space diagram changed into an helix. The idea of flow is, even to physicists, a nearly mystical concept but generally flow is said to be either laminar (smooth) or turbulent. (I am going to ignore the transit between these states for now.) So I am saying that the mind's activity on any good day is a dynamic system interacting, now turbulent now laminar all depending on the number of dimensions (degrees of freedom) in which it is operating. (Note freedom here does not pertain to the various rhetorical machinations of classical "Philosophy" but rather to the directions, trains of thought or what have you that the mind is able to move in.) It will quickly be seen that the more degrees of freedom that a mind has the more diffuse and non focused it is. Thus the greatest freedom of the mind is not only its largest dimension but also its least focused. We might make the analogy of a mind in this diffuse state to a waterfall whose complexity of flow defies calculation, at lest in the classical way of looking at that kind of system. The meditative state is a very different kind of system. In the meditative state many of the dimensions of the wandering mind as discussed above have been eliminated.

Now that we have connected the idea of a dynamic system to the mind it might be interesting to draw a few analogies. One interesting analogy is that from the relative chaos of the untrained mind learning to limit the degrees of freedom of the system produces meditative states like the one described in the section on Raja yoga. Another interesting analogy can be made by considering the development of consciousness from infancy. During this development an infant's mind passes from diffuse states (turbulence) to ordered states (laminar) only to disappears again into chaos. This disappearance and reemergence of ordered states characterizes the learning process that human beings go through on all scales. This kind of behavior is very indicative of the fractal dimensional model of brain activity. Whenever we see something creating self similar patterns on many scales of measure then we can be very confident that we have found a system functioning in a fractal dimension. The resolution of chaos back to laminar flow would seem to negate some of Prigogines results in his studies of the arrow of time (the non reversibility of chaotic systems) at least for systems like the human brain. For example: the mind may start out in a laminar flow pattern as a very young child but that state quickly gives way to turbulence as the child reaches pre-teen stage in its development. This turbulence is the result of the maturation process as more and more of the senses become active. Once this process has reached a critical level (Feigenbaum's constant perhaps) turbulence sets in and the system becomes chaotic. From this chaos develops all of the normal mental patterns, the personality, memory and so on that we regularly see manifested in human beings. But why should this be so? Why should development continue on one course or another why doesn't a child who develops alongside other animals, dogs, cats, birds, insects begin to develop roach like characteristics or dog like characteristics? Well.... everyone knows the answer to that one right? Genetically we have a pattern which causes certain features to develop and not other features. That pattern, DNA, is self similar, bounded and yet infinite in the sense that there are a certain number of chromosomes, thus the system is bounded, but from these chromosomes can come an infinite number of personalities. OK we will say that the above situation fairly well accounts for things like hands and feet and noses (although it doesn't really) but what about behaviors like grooming or barking or hiding in dark places? Well in fact children often display non-human behaviors and these behaviors are unusually not rewarded by the parents and thus are short lived transients in the overall pattern of the personality. What is it that ties this infinite diffuse state together with the focused meditative state? What kind of system, concept, archetype or god can resolve this dichotomy? But I am getting a little ahead of myself here first I need to show that there are patterns in chaos. This is no easy task as many books have been written on the subject but I will append a set of drawings from "Chaos Making of a New Science" by James Gleick. These drawings show a Strange Attractor system in several states of development:

The knot mass of the 3 dimensional representation of this strange attractor does not appear to have much of a pattern but once we take a look inside that mass we find a beautiful folding of space. This 2 dimensional sample or slice of the 3 dimensional representation shows us the inside of the attractor orbit. There is definitely a pattern here a pattern in chaos. I am going to leave this subject now and let the curious explore the many books and essays written upon this subject. I will provide a brief list of suggested further reading on the subject at the end of this text.

Now that we have determined that there is a definite pattern in even the most chaotic system we will more readily understand why no two children develop in exactly the same way. Although there may be an infinite number of possibilities for development of the basic model known as human, the Archetypal Human being, in the role of the Attractor, pulls the system into its own pattern. This pattern is what enables one person to recognizeanother person as human. Even though beyond that basic pattern - bi-pedal, stands erect, generally uses utensil - there is precious little similarity (i.e. chaos) inherent in the idea of "human". This pull is very strong and thus even major aberrations in the path of an individual can be corrected for and the individual brought back into the basin of attraction. These attractors are very probably the actual Archetypes spoken of by Jung but never fully understood by him.

If an individual human being decides to push further than this normal turbulent flow then he or she must again reach a state of laminar flow. From this new, or regained, state of laminar flow a new level or scale is reached and at this new scale another "chaos" develops, another Strange attractor, and this is what the mystics have called enlightenment. Thus these states of laminar flow are possibly only bridges to various levels of scale of self similar attractors. These "Smooth Points" connect one level of scale with another. The "Chaotic" or attractor state of an individual would then be thought of as being the normal functioning mode of the individual. So the question becomes are we seeking a reversal of time's arrow or are we seeing self similarity as we jump from scale to scale? This is a fascinating area for study and this little diversion gives us an example of that richness.

Back to the Raja yoga study we see that the erratic jumps in the amount of breaks, resolved themselves into an oscillating pattern. Although the above example does not go beyond oscillation, further experiments produced an oscillation which steadily decreased in amplitude. This clearly indicated that the mind had reached a laminar flow. From this laminar flow the magician reached new understandings of his being and was assisted along his path by these new understandings.

The training produced:

1. A new state of the attractor which is this magician's being.
2. A pathway whereby he can slip into laminar flow at will anytime a new scale (a new perspective, a new understanding) is needed.

According to the technical language of the Hindus', the magician in the above example reached Dhyana through this process and was thereby informed of Samadhi through subsequent experiments and practices. The Raja Yoga led to the knowledge of the way to transit the states of laminar flow which in turn led to the various manifestations of the Attractor as the being of the magician.

In this text I have tried to point out some areas of similarity between meditative states, the science of number as it is beginning to be formed by discussions of fractals, chaos, attractors and so on and various aspects of human psychological development. This is of course an enormously broad topic. No one could reasonably hope to do more than excite further study and research on the part of the audience. I hope that I have excited just such a desire within some of the audience members here today. I feel that to search this vast and rich realm for the keys to the marriage of the science of the profane and Our science may well be the most important task that any of us can perform in this first part of the Aeon of Horus. It is my belief that a marriage of "their" science and Our science will benefit both researches immensely as ultimately, I believe, we will find them one and the same.

Selected Bibliography of suggested reading:

Turbulent Mirror by John Briggs and F. David Peat
The Mathematical Tourist by Ivars Peterson
Order Out of Chaos by Ilya Progogine and Isabelle Strengers
Strange Attractors by Julien C. Sprott
Chaos Making of a new Science by James Gleick

The foregoing is by no means an exhaustive list but it will serve to get a student started in his or her explorations of this fascinating area.

Note on this essay: The original essay was given as a lecture at the Third annual HOOR convention in Louisville KY, June 15th 1996e.v. This present version is revised.