Ernest Lawrence Rossi

In exploring the possibilities for creating a model of the creative cosmos, it is important to seek mathematical models that can express the common relationships unifying the sciences of matter, life and mind. Current candidates for such all embracing mathematical models come from the burgeoning fields of non-linear dynamics that are described under varying labels as Chaos Theory, Self-Organization Theory and Adaptive Complexity Theory. In this paper we will limit ourselves to only one mathematical model called, "The Feigenbaum Scenario." A major virtue of The Feigenbaum Scenario is that it is very easy to understand by non-mathematicians and it has led to new depths of profound understanding in a wide variety of the physical and biological sciences. Everything from purely mechanical systems, fluid dynamics and the weather to the patterns of biological growth in nature and the dynamics of heart, hormone and brain rhythms have been found to exhibit aspects of the Feigenbaum scenario. We explore the possibility that the Feigenbaum scenario can be extended to experiences of mind, sensation, perception and human behavior as well. We conclude that a major function of consciousness may be to transform the non-linear, irrational, unconscious and difficult to predict dynamics of unconscious nature into the more linear, rational and predictable psychodynamics that make human experience and social life possible.

Key words: Feigenbaum scenario, chaos theory, psychoanalysis, mind, behavior

In his book on The Creative Cosmos Laszlo (1993) outlined the goal of a unified science of matter, life and mind with these words.

"Binding together the observed facts in the simplest possible scheme is a perennial goal of systematic thought in science as well as philosophy. It is also the goal of this study. We attempt to elucidate the unified interactive dynamics (UID) through which the facts investigated in physics, biology, and the sciences of mind and consciousness could be simply and coherently bound together." Laszlo (1993, p. 134)

The most interesting current candidates for binding together the basic sciences of physics, biology and psychology are to be found in the so-called "New non-linear dynamics of Chaos Theory." The source of present day investigations of non-linear dynamics can be traced to the turn of the century French mathematician, Henri Poincaré (1905/1952). Poincaré developed his mathematical ideas of non-linear dynamics to deal with deep problems in physics at the same time that Sigmund Freud and Carl Jung were formulating the foundations of psycho-dynamics to deal with deep problems in human psychology. Until now there has been no bridge built between the ideas of the mathematician and the psychologists. A recent investigation of the similarity between the concepts of the non-linear dynamics of Chaos Theory and the psycho-dynamics of depth psychology, however, suggests they may share a common conceptual foundation (Rossi, 1996).

This paper first outlines the concepts of linear and non-linear dynamics, Chaos theory and the Feigenbaum Scenario. Research that suggests how the Feigenbaum Scenario could be used as a mathematical model of data in sensory-perceptual psychology will then be presented. We will explore the mental and behavioral phenomenology of depth psychology that could be modeled by recent developments in Poincare’s non-linear mathematical dynamics and the Feigenbaum Scenario. The implications of this association between psycho-dynamics of Freud and Jung and modern chaos theory for creating a new informational approach to unified interactive dynamics (UID) of matter, life and mind will then be discussed.

The typical linear approach to analyzing data in psychology is illustrated on the left in figure one. The straight line cutting through a cloud of data points is the "best linear fit," or the "best statistical approximation" of the whole cloud. Traditionally each of the data points that make up the cloud is said to be a combination of a real psychological factor and experimental error or "noise."

The non-linear dynamics systems approach, however, proposes that the apparently random deviations of many points around the straight line actually may be the signature of chaos when studied over time. Psychological factors over which we do not have complete control could be responsible for deviations from the straight line that are called "Noise" (Combs and Winkler, 1995). The recent re-examination of the field of classical psychophysics -- the relationships between physical stimuli and human perception such as how bright a light appears to be -- confirms this. Current researchers find that the noise, once thought to be due to errors of measurement, may actually be an important part of the non-linear dynamics of perception and cognition in self-organizing our view of the world (Gregson, 1992; Guastello, 1995).

The non-linear relationship between performance and arousal or anxiety illustrated on the right of figure one, called the Yerkes-Dodson function (1908), is an example of one the earliest and most well established laws in psychology. Guastello (1995) has updated the significance of this relationship in terms of Thom’s (1972) catastrophe theory of non-linear dynamics in social organizations and the psychosocial factors in accidents and the work place. This intensively researched relationship between arousal and performance is an important foundation for a new concept of the mindbody relationship in psychotherapeutic work (Rossi, 1996).

One way of understanding Self-Organized Dynamical Systems is to recognize how they are made up of one or more parts that can "communicate" with each other with feedback loops as illustrated in Figure two. These feedback loops form a kind of reciprocal or circular causation which is the fundamental process in the shift in current interest from linear to non-linear dynamics in psychology.  The nested circular loops in Figure illustrate how the classical 4-stage cycle of creativity in individual experience (inner circle) corresponds to the 4-stage break-out heuristic in psychotherapy (middle circle: (Rossi, 1968, 2000, 2002), and the 4-stage cycle of the monomyth of the hero's journey (Campbell, 1956; Rossi, 2000).  The outer labels in Figure 2 indicate the corresponding 4-stage process in the psychosocial dynamics of political identity.

Fig. 2 The 4-stage cycle of creativity first conceptualized by Poincare (inner circle), psychotherapy (middle circle), the journey of the hero (outer circle), and psychosocial identity in the corresponding 4 stages of the political process (outer labels) (From Rossi, 1968, 2000, 2002). 

Virtually all  dynamical systems, life as we know it, are made up of multiple feedback loops and non-linear patterns of circular causation as illustrated in figure 2 on all levels from  gene to mind and psychosocial processes . This makes it very difficult to untangle the simple cause-effect relationships, that are the ideal of the older classical linear mathematics, in the life sciences and psychology. This is why we have such puzzlement about cause and effect in psychology: Which came first the chicken or the egg? In fact, now we can say that whenever we are confronted with such a circular causation puzzle or paradox, we are entering the area of non-linear dynamics! Even if we cannot entirely sort out, predict and control these circular dynamics, the non-linear approach usually gives us a more realistic picture of the inherent individuality of complex adaptive systems such as human personality and social structure. Above all, the Complex Adaptive Systems approach to Self-Organization helps us avoid the tragic illusion that the exercise of arbitrary direct suggestion, programming, manipulation and control over humans is an option in personal, professional and social relationships. It is usually the found that the use of political policy to directly manipulate human behavior leads to non-linear effects that are almost impossible to predict and control (Vallacher & Nowak, 1994). The implications such non-linear dynamics for any ethical theory designed to facilitate the evolution of optimal human development and psychosocial processes open profoundly new vistas for social science and philosophy.

The recent fascination with the mathematics of non-linear dynamics that leads from order to chaos has its source in the fact that a very simple equation can have more than one answer. That is, some equations, as most of us found to our confusion in high school, have multiple solutions. Solving the simple equation x² = 4, for example, produces two solutions: x = 2 and x = -2. In figure three we illustrate this bifurcation of solutions in the exploration of a similar quadratic equation in a process commonly called feedback, recursion, or iteration. While each of these three terms have different connotations, we use them here to describe a basic mathematical operation that is important for modeling the common dynamics of matter, life and mind. The basic process for generating the Feigenbaum Scenario is to take the answer from the first equation and feed it back into the same equation to get a new answer and keep doing this over and over again. Note that this is the same fundamental process that takes place in all biological and psychosocial systems: life at all levels from the molecular-genetic to the psychosocial involves doing the same operation over and over again in the process of evolutionary adaptation as well as daily survival. At the molecular-genetic level, for example, gene transcription and translation takes place over and over again in adaptation to feedback from the environment. In psychology one stimulus-response unit of behavior is feedback for the next; the experience of one thought becomes feedback leading to the next etc. The Feigenbaum Scenario is thus an unusually appropriate model for the life sciences because it models the actual archetypal process of feedback that is used in all living systems rather than simply giving us a correct answer or prediction (regardless of how differently the process of how the model and life systems actually operate).

Is there any lawfulness in this mathematical feedback process of the Feigenbaum Scenario? Are there any patterns in the series of feedback or iterated answers it generates or are they random? The surprising answer demonstrated by Mitchell Feigenbaum in 1975, while he was still a graduate student in physics, is that both order and chaos are generated in the pattern of answers that come from this simple process of iteration. That is, from this mathematical point of view, order and chaos are not opposites as is commonly supposed. They are both stages of a more general process of modeling complexity or self-organization in numbers and nature when the same feedback operation takes place over and over again.

Even more surprising was the discovery that there is a well defined path or route which leads from order to chaos that is described as "universal" (Feigenbaum, 1980). It is universal because the same abrupt changes between order and chaos, usually called "bifurcations," can be found in many apparently different equations used to model different processes in nature. When the series of iterated answers to different equations are graphed they all have features in common with the Feigenbaum diagram illustrated in figure three.   Peitgen et al. (1992, p. 587) have said, "The Feigenbaum diagram has become the most important icon of chaos theory. It will most likely be an image which will remain as a landmark of the scientific progress of this century."

The word bifurcation, then, simply means a sudden change in the pattern or number of solutions to an equation as a parameter is varied. In the equation below the letter "a" represents a parameter that acts as a kind of control valve on the expression of the equation. The value of the parameter at which the bifurcation takes place is called, logically enough, a bifurcation point or bifurcation parameter. A bifurcation diagram is made by plotting a parameter on one axis and an important variable on the other. The essential dynamics of a Feigenbaum bifurcation diagram is illustrated by a branching tree in figure two where each branch represents an answer or "choice" in the series of solutions to an equation obtained by a process of feedback or iteration. Choice? Mathematical bifurcation is a natural consequence of the way numbers work with feedback or iteration. This model helps us understand how many physical and chemical systems in nature and virtually all complex biological and psychological systems involve multiple processes of feedback. Since mind and behavior obviously utilize information feedback on many levels we naturally wonder whether such bifurcation models can illustrate anything interesting about human choice points on a conscious and/or unconscious levels.

That is the controversial question that we would like to explore with one of the most well known equations used to demonstrate non-linear dynamics: the so-called logistic equation that was originally proposed as a model of population dynamics where feedback prevents populations of bacteria, plants and animals from growing infinitely because of environmental limitations of food supplies and space as well as the presence of predators. Can the logistic equation also be used to illustrate any facets of the population dynamics of ideas, awareness or consciousness? In the logistic equation

x₁ = a x ₀ (1 - x₀ )

the initial value (x₀) is feed back into the equation to get first solution (x₁). This first solution is then feed back into the equation to get the second solution x₂ as shown below.

x₂ = a x₁ (1 - x₁ )

Continuing this feed back process leads us to a series of solutions that are illustrated in the bifurcation diagram illustrated in figure two. The first long stem coming down from the top of figure two represents a series of solutions that then branches or "bifurcates" as indicated and from each of these two branches we see two more bifurcating again and so on. This is called the "period-doubling regime" of the Feigenbaum Scenario on the path from order to chaos. Notice that the branches get shorter and shorter as they move down until a threshold is finally reached, that is now called the Feigenbaum point, after the fourth bifurcation (so small it is too difficult to see and label in figure two) where the system falls into chaos illustrated as the dark but structured smudge. There is a ratio that quantifies the period doubling path to chaos that is found to be true of many different equations when they are iterated. This ratio is called Feigenbaum’s Constant which converges to a value of 4.6692... This ratio is obtained is made up of the lengths of any two successive branches. The Feigenbaum’s Constant is now regarded as important in dynamics theory as the number Pi is to geometry.

While the Feigenbaum point marks the onset of "deterministic chaos," it is not really random from a statistical point of view. Deterministic Chaos only looks random because of the limitations of human perception. If we zoom in on any small portion of the diagram with a computer we will find that the overall picture is reproduced again in the smaller portion that we blow up. This is called the "fractal" or "self-similar" aspect of the equation on all scales. Notice that when we look at the diagram we can easily see about seven paths clearly as labeled. If you are really talented you might spot more, as many as 15, if you count the next lower bifurcating level carefully. But after that the distribution of paths seems to be a chaotic smudge with some vague structure and blank spaces from about the middle to the bottom of the diagram. Kihlstrom (1980) found that 15 items was the upper limit for post-hypnotic memory in low as well as highly susceptible hypnotic subjects but they achieved that level of performance differently.

A number of classical studies in psychology confirm that seven units (plus or minus two) is, in fact, the usual limit of human perception (Miller, 1956). Sperling (1960), for example, found that people could remember about seven letters over a 1-second interval. He called this the iconic trace. Neisser (1967) found that the auditory trance, which he called echoic memory, had similar characteristics. Recall how telephone numbers in most developed areas also have seven digits. Is the number seven as a band-width in human sensory-perceptual studies and the seven paths we can see easily in the bifurcation diagram simply a coincidence? Or is it another example of the seemingly unreasonable effectiveness of mathematics in modeling human experience in the most unexpected ways? Research extended to other sensory and perceptual levels of human experiencing is now needed to confirm the relevance of the number seven and the Feigenbaum point in human awareness. Freeman’s (1995) research on the non-linear dynamics of the sensory-perceptual dynamics of smell, for example, would be an important test case.

Does the birfurcating Feigenbaum diagram illustrate anything else of interest about human awareness and perhaps conscious-unconscious dynamics in general? Notice the "bubbles" that appear in the bifurcating diagram of figure four for the more complicated two variable equations of the Henon system

(x, y) ---> (1.25 - x² + ay, x).

Quite apart from the mathematical rational for these bubbles which is beyond the scope of this paper, why do such bubbles appear in the Feigenbaum feedback process of iterating the Henon system? To continue our metaphor, are these bubbles "islands of awareness" surrounded by deterministic chaos that human perception is too limited to discern clearly? Do the bubbles represent a few groups or levels of the laws of nature that we can discern in the vaster darkness that surrounds human understanding. The physicist John Archibald Wheeler (1994) has described such limitations in our perceptions of the laws of nature and how we can cope with them. Any "law of nature" is not really out there, it is simply our human short-hand way of summarizing a little bit of our perception of nature. The pioneering depth psychologist Carl Jung has described "islands of consciousness within the unconscious" that seem to be modeled by these bubbles. One can only wonder whether these bubbles are akin to dreams, fantasies and creative inspirations that seem to bubble up spontaneously from the unconscious.

Ernest Hilgard (Hilgard and Hilgard, 1983) reports experimental research in hypnosis that provides evidence for a "hidden observer" that is able to verbally report on what is being experienced (e.g. pain) on a deeper level even when the conscious personality does not report the experience of pain. Is this hidden observer and, perhaps more generally, the dissociated experiences of multiple personality, being illustrated in the bubbles in the birfuraction diagrams? Is not pain and, indeed, all "symptoms" to be understood as forms of information, messages or awareness that bubbles up from the unconscious (deterministically chaotic) levels of the body?

Has consciousness itself evolved as a practical method of coping with the deterministic chaos of living experience? Could this be part of the answer to the "hard question" of why consciousness has evolved (see the special issue of the Journal of Consciousness Studies, vol. 2, 3, 1995 devoted to "Explaining Consciousness -- The Hard Problem). Consciousness, according to this intuition, is a sort of amplifying lens or mirror that allows us to sort out or focus various aspects of non-linear experience that are necessary for survival.

At present we can only speculate about whether the significance of the number seven in gambling and many mantic and mystical belief systems could have the same source in the limitations in our sensory-perceptual-cognitive awareness that may be illustrated in the bifurcation diagrams. Seven items are about as much as we usually can hold in consciousness so we feel we know them and we are comfortable with them. When consciousness has to juggle more than seven items, dimensions or levels, understanding seems to become chaotic, dark, fearful, unconscious and perhaps unreal though we may have dim intuitions of other levels that seem to be in the realm of prophesy. What are the mathematical implications and predictions of the bifurcation diagrams regarding human experience that could be tested experimentally? The problem is that apart from the types of intuitions reported here, at the present time the connection between the bifurcating diagrams and human experience is not well understood but remains an area of pioneering research (Rössler 1992 a &b; Vallacher and Nowak, 1994; Guastello, 1995).

The universality in the appearance of the Feigenbaum numbers in many complex systems of a different nature allows the prediction of the onset of turbulence in dripping faucets as well as torrential rivers, the oscillations of liquid helium, electric circuits and the fluctuations of insect and animal populations. There are a number of fascinating though highly speculative views about the possible significance of the Feigenbaum Point for psychology, sociology and the humanities in general. Merry (1995, p. 37) suggests, for example, that the Feigenbaum Point is where systems cascade into chaos "where infinite choices create a situation in which freedom has no more meaning."

Could we generalize this to say that emotions, imagery, behavior and cognition and, yes, even psychosomatic symptoms that have lost their meaning have somehow fallen into the chaotic regime within "experiential space" where even our sense of reality teeters off the edge of understanding or rationality? Does this suggest that beyond the Feigenbaum Point inner experience may fall into a sense of what we call "unreality?" Put another way, does the Feigenbaum point signal the division between primary process (irrational) versus the secondary processes (rational, ego processes) as defined in psychoanalysis? In this sense, would the Feigenbaum point also represent the limit of our sense of voluntary ego control over our mental experience and behavior? If a highly hypnotizable subjects report a sense of involuntaryness in their experience and behavior does that mean they have moved into the chaotic realm? The physicist uses the route to chaos as a way of describing turbulence in nature (fast moving water flowing over rocks, air turbulence behind an airplane etc.). Do we have another analogy here between physics and psychology by saying that the experience of confusion or disorientation is the turbulence of the mind moving past the Feigenbaum point into chaos?

How would we actually measure such a Feigenbaum Points of inner experience? We could go on and on with such questions that are intuitively provocative but which we at present do not know how to answer. Until we develop new methodologies to test hypotheses about the relationship between the mathematical formalisms of chaos theory and human experience we really do not have a new science of the dynamics of human nature. We still do not know whether the Feigenbaum Point is simply a provocative metaphor for psychology or a major breakthrough into the possibility of formulating new mathematical models with conceptual as well as predictive power.

Does the chaotic regime of the brain-mind make up what we call the "unconscious?" Is that why during those brief moments of introspection as we are falling asleep or awakening we occasionally glimpse what from our conscious perspective seems to be a confused plethora of inchoate images, feelings, thoughts and what not? Are there circumstances when consciousness gets caught in the chaotic realm so that the person experiences a "dissociation" and/or a sense of "identity loss?" Is this what some cultural and spiritual traditions call a "loss of soul?" Obviously such questions will be a rich area for exploring new ways of reconceptualizing the foundation of human consciousness and psychological experience in a unified interactive dynamics that shares the same universal principles that govern other complex systems in mathematics, physics, biology and ecology.

Another fascinating historical parallel between the psychodynamics of depth psychology and the nonlinear dynamics of Chaos Theory is in their efforts to bridge the great polarities of human experience. These polarities have been described in a variety of ways in different cultural traditions as: the Conscious vs. Unconscious; the Rational vs. Irrational; the Apollonian vs. Dionysian; the Yang vs. Yin; and now the Linear vs. Non-linear.

In Freud’s psychodynamics the relation between the conscious and unconscious, the ego and the id, was epitomized by the psychoanalytic mechanisms of defense such as rationalization, repression, displacement, projection and sublimation. For Jung the relationship between the conscious and unconscious was described as the transcendent function defined in this way.

"There is nothing mysterious or metaphysical about the term ‘transcendent function.’ It means a psychological function comparable in its way to a mathematical function of the same name, which is a function of real and imaginary numbers. The psychological ‘transcendent function’ arises from the union of conscious and unconscious contents." (Jung, 1960, p. 69).

For Jung the transcendent function was mediated by a special form of self-confrontation via inner imagery and dialogue that he called "Active Imagination." Prior to these efforts from depth psychology were the labors of philosophers and logicians such as George Boole, who laid the groundwork for modern symbolic logic with his celebrated treatise "An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probability," (Boole, 1854/1958). Boole formulated the laws of the linear and rational thought as they are perceived in conscious human experience. What he left out was the nonlinear and irrational of human experience, precisely what is of most interest at the present time in portraying the emergence of the self-organizing processes of nature and mind. Our focus on nonlinear dynamics is thus an approach to formulating the laws of the irrational in human experience to complement Boole’s focus on the laws of rational logic.

I propose that the non-linear dynamics of human experience are most readily observed in transition between the unconscious and consciousness that we describe as the dreams (rapid eye-movement sleep) at night as well as the semi-autonomous experience of fantasy when we are awake. I outlined the subjective experiences of psychological growth and signs of the evolution of consciousness in dreams in about two dozen hypotheses that are described and illustrated with clinical case material elsewhere (Rossi, 1972/1986/1998, 2002). More recently a number of investigators have developed new systems of bridging the gap between the rational and irrational on the borderline between psychology and nonlinear science. Shawe-Taylor (1996), for example, have described consciousness as a linear phenomenon. From our perspective their theory provides an opportunity to deepen our understanding of at least one major function of consciousness: mediating the transition between the nonlinear and the linear. It is close to what Freud called "rationalization." The Shawe-Taylor theory is that the essential function of consciousness is to transform the nonlinear dynamics of the unconscious into a linear conscious system of relationships.

Why has this transformation from the non-linear to the linear evolved in our consciousness of nature? Recall that the long term behavior of feedback in non-linear systems (the realm of chaos in figures 3 and 4) is very difficult or impossible to predict. Consciousness has the Herculean task of remapping, transducing, transforming or reframing, if you will, the natural non-linear dynamics of our unconscious physical and biological nature into the linear, rational and at least somewhat more predictable dynamics of our subjective experience of nature.

Consciousness, of course, is only partially successful in this heroic task. As we all know, we are continuously engaged in an unceasing Sisyphean struggle to make sense of the world. It is all to easy to fall into the irrational and chaos in the negative sense of unpredictability. What is most interesting about the Shawe-Taylor theory of the linearization function of consciousness is the evidence they marshal for it from research on (1) artificial neural networks in neuroscience and (2) Aaron Beck’s (1976) version of Cognitive Therapy. In brief they conceptualize Beck’s Cognitive Therapy as an exercise in facilitating the linearization function of consciousness in correcting the errors and unpredictability of the nonlinear dynamics of nature and mind.

Fig 5 The linearization function of consciousness as proposed by Shawe-Taylor (1996).  There are a number of conceptual or inferential errors found in psychopathology.  Over-generalization, for example, is the process of using a single incident to draw general conclusions considered operative over all situations.  Personalization is the error of identifying external events with oneself when there is no objective basis for making such associations.  Magnification and minimization refer to the bifurcating assessment of events, skills, etc.  For example, a person may magnify the difficulty of a task and minimize his ability to deal with it.  Arbitrary Inference (not shown) is a process of drawing a conclusion without sufficient evidence to support it.  How to correct such "errors" of the natural non-linear dynamics of cognition is one of the most important basic issues distinguishing the various schools of psychotherapy.

The linearization function of consciousness by Shawe-Taylor (1996) is illustrated in the figure five where conceptual graphs of the non-linear "errors" the mind is prone to are presented. Why did the linearizartion or rational function of consciousness evolve? Nonlinear processes are usually difficult and, indeed, most often impossible to predict. I therefore speculate that consciousness evolved to linearize bounded bits and pieces of nonlinear nature so that the organism could predict important ranges of experience necessary for survival. Linearization is involved in transducing the non-linear universe into sufficiently predictable patterns - just predictable enough to insure survival. A prey animal like a deer or rabbit, for example, has to evolve just enough linear sensory-perceptual-motor organization to predict and avoid the trajectory of a predator pursuing it. Just as the mathematical process of linearization is always bounded within certain narrow limits in which it is effective, so rational consciousness is also bounded in its linearization and capacity to predict non-linear nature. When we attempt to step beyond the narrow limits of the linearization function of our current level of conscious we fall into conundrums and paradox. This leads to the deep epistemological speculation that the boundary conditions of the linear, rational world view of western consciousness are being outlined by many of the most famous paradoxes of our century: Bertran Russell’s paradoxes of logic and Gödel’s Incompleteness Theorem, Turing’s Halting Problem, the Uncertainty Principle in quantum physics and the dynamics of chaos in the Feigenbaum Scenario of a unified science of matter, life and mind.

The Feigenbaum scenario is presented as a mathematical model of the creative cosmos that could express the common dynamics unifying the sciences of matter, life and mind. The deep correspondence between the new non-linear dynamics of the Feigenbaum Scenario, Chaos Theory and the classical psycho-dynamics of historical hypnosis, psychoanalysis and that suggest they all derive from a common archetypal foundation. The new sciences of Self-Organization and Adaptive Complexity model human nature and consciousness as a non-linear dynamic of ever-shifting states evolving on the critical edge of deterministic chaos. A major function of consciousness may be to transform the non-linear, irrational, unconscious and difficult to predict dynamics of unconscious nature into the more linear, rational and predictable psychodynamics that make human experience and social life possible.

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