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1993 27: 16J Spec EducDoug Guess and Wayne Sailor

and Developmental DisabilitiesChaos Theory and the Study of Human Behavior: Implications for Special Education

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THE JOURNAL OF SPECIAL EDUCATION VOL 27/NO. 1/1993/pp. 16-34

CHAOS THEORY AND THE STUDY OF HUMAN BEHAVIOR: IMPLICATIONS FOR SPECIAL EDUCATION

AND DEVELOPMENTAL DISABILITIES

Doug Guess Wayne Sailor

University of Kansas

An introduction to the concepts and terminology many diverse fields and disciplines, such as mathe-of chaos science, a contemporary approach to sys- matics, physics, meteorology, economics, biology, tern theory, is provided, along with its implications chemistry, and psychology. This paper presents for the behavioral and social sciences and, espe- some basic tenets of chaos science, defines a num-cially, the field of developmental disabilities. ber of the key concepts of chaos modeling that Chaos is a dynamic, macroanalytic approach to hold particular relevance to the social sciences, understanding interactive components of complex and discusses applications of chaos science to spe-systems. Chaos represents a rapidly growing body cial education and the field of developmental dis-of knowledge that has attracted the attention of abilities.

The behavioral and social sciences are witnessing increasing interest in models of scientific investigation that are focused on the dynamics of whole systems (Horowitz, 1987; Schwartz & Ogilvy, 1979). This trend (anticipated over two decades ago by Bertalanffy, 1968) emphasizes a macroanalytic approach to understand-ing behavior wherein phenomena that comprise components of a larger system are assumed to be interactively related in a nonlinear fashion. Two major macro-analytic investigative models have begun to assert themselves in recent years. One is the qualitative research approach, with its epistemology grounded in relativism and phenomenological inference (e.g., Lincoln & Guba, 1985). The second is system theory (Yates, 1987), an example of which is the subject of this paper—chaos science (e.g., Gleick, 1987). The explosion of knowledge that has thus far accom-panied chaos science (Percival, 1989) affords an opportunity to challenge and stimulate the social and behavioral sciences, including developmental disabilities; to chart new directions in the research conducted; to structure some of the kinds of questions to be asked; and to provide the eventual transition of empirical find-ings to the level of practical application.

Chaos science is a systems approach that now appears in the literature of numer-ous fields of inquiry, with a variety of rubrics. In geological science, for example, Bak and Chen (1991) have employed the principles of chaotic modeling to the

Address: Doug Guess, Department of Special Education, 3001 Dole, University of Kansas, Lawrence KS 66045.

16



THE JOURNAL OF SPECIAL EDUCATION VOL. 27/NO. 1/1993 17

study of earthquake prediction under the term "self-organized criticality." Schoner and Kelso (1988) in neurophysiology and Thelen (1990) in developmental psy-chology have referred to "dynamical systems," whereas biologists Prigogine and Stengers (1984) and Yates (1987) have written of "self-organizing systems." The emergence of the term chaos to characterize the broader class of cross-disciplinary systems models and theories was popularized by Gleick (1987), although Stewart (1989) attributed the initial demarcation of chaos science to the mathematician Henri Poincare in the late 1800s. Our purpose in this paper is to provide an intro-duction for special educators to chaos science and to show how some of the under-lying concepts have direct relevance to the study and analysis of human learning, including, in particular, persons with disabilities.

Chaotic Behavior

Gleick (1987) posited that chaos, relativity theory, and quantum mechanics represent the major scientific advances of the 20th century. Within the last decade, derivations of chaotic phenomena have been intensely studied in such diverse sciences as fluid dynamics, genetic variation, population changes and predictions, and economics (Coveney, 1990). Chaotic behavior has been investigated and dis-cussed in relation to electroencephalographic patterns (Basar, 1990), heart rhythms (May, 1989; Pool, 1989b), the physiology of olfactory perception (Freeman, 1991), social interactions during group therapy sessions (Lichtenberg & Knox, 1991), childhood disease epidemiology (Olsen & Schaffer, 1990), and ecosystems (Bak & Chen, 1991). Gleick (1987) observed that chaos has helped to reverse the trend toward specialization among disciplines by offering universal applicability of prin-ciples and concepts across varying areas of scientific inquiry, at both basic and applied levels. This type of "wholeness" also supports the general system theory of Bertalanffy (1968), who earlier anticipated the presence of isomorphic laws and principles across disciplines.

Chaos may best be understood as a dynamic view of phenomena that represents in the behavior of systems a midpoint between strict determinism and total ran-domness. Its primary point of application occurs when the condition of a system changes over time, although the term chaos is a bit of a misnomer. Conventional wisdom would associate the term with complete randomness in the state of a sys-tem, without possibility of prediction and control. Chaos science, however, is based on a more complex set of assumptions. On the one hand, the "noise" in a system (unexplained variance) can be shown to have a certain degree of predictability, particularly when the parameters of an open system are expanded. On the other hand, chaos implies that total prediction for many phenomena can never be achieved, no matter how precise are the measurement systems used (Pool, 1989a). As observed by Crutchfield, Farmer, Packard, and Shaw (1986), "chaos provides a mechanism that allows for free will within a world governed by deterministic laws" (p. 57).

An example of weather prediction exemplifies this concept (Gleick, 1987). Long-term global patterns of weather can be predicted with a degree of certainty, as exemplified by computer models of droughts associated with changes in seasons



18 THE JOURNAL OF SPECIAL EDUCATION VOL. 27/NO. 1/1993

of the year. Additionally, most 1- to 2-day forecasts are fairly accurate. It is extremely difficult, however, to accurately predict specific weather patterns for any particular location over a longer time span (e.g., several weeks in advance). This difficulty was described earlier by meteorologist Lorenz (Gleick, 1987), who demonstrated that only a tiny incremental change in a weather computer predic-tion program resulted in a radically revised forecast outcome. This phenomenon is referred to as sensitive dependence on initial conditions. It was initially called the "butterfly effect" to make an exaggerated point that the flapping of a butterfly's wings might set into play a chain of events that can so affect a global weather system that accurate prediction is rendered impossible. The result, chaotic behav-ior, is thus produced when the multiple "trajectories" of various components start very close together and over time diverge rapidly as an outgrowth of sensitivity to the initial conditions of change in a system (Holden, 1985).

Thelen (1990) used, in part, the phenomenon of sensitive dependence on ini-tial conditions to explain individual differences in humans. Whereas major devel-opmental progressions can be predicted, individual differences result from the sensitive influences of a variety of unpredictable endogenous and exogenous fac-tors occurring during early infancy and childhood. Thelen rejected linear models of quantitative behavioral genetics that argue that all of the relevant variance in the study of individual differences can be estimated through measures of spe-cifiable heritable and environmental factors in linear fashion (Plomin & Daniels, 1987). Exogenous factors that may influence development in a compound manner over time ("stochastic events") are discussed as "a dead end for research" (Plomin & Daniels, 1987, p. 8; cited in Thelen, 1990, p. 20). In Thelen's view, linear models are context bound and unidimensional. Human differences are better understood through the study of "open systems" (dynamical models), wherein constrained (predictable) influences interact in time with random influences. According to Thelen, the intrinsic "noisiness" (as in "white noise") of a system serves both as the wellspring of its stability and, in part, the determiner of its total pattern, but it also serves as the determiner of its variability when viewed from the standpoint of the end state of the trajectory of the components studied. Human variability thus may be demonstrated to arise from "the intricacies of ontogeny alone" (Thelen, 1990, p. 28), and can never be wholly accounted for by any linear deter-ministic heredity/environmental model.

The implications of large-scale changes resulting from change influences in ontogeny result in a radically altered view of causation in differential psychology. Chance interventions can be viewed not merely as a source of variability in linear dynamics, but as an inevitable consequence of human ontogeny and as "the very pool from which change is sculpted" (Thelen, 1990, p. 38). Chaos theory predicts that there will be times when an open system will be highly variable and thus crit-ically sensitive to perturbations or external influences. According to Thelen (1990), if one conducts a careful examination of individual trajectories in a longitudinal design, it may be possible to capture the influence of critical events, as viewed from the endpoints of the trajectories during the period of initial sensitivity to these influences. When such influences can be examined across systems or across individuals (or components) within systems, then the unexplainable (unpredict-



THE JOURNAL OF SPECIAL EDUCATION VOL 27/NO. 1/1993 19

able variability) has the potential to join the ranks of the explained. Chaos science is the search for influences during periods of sensitivity to initial changes in com-ponent trajectories within open systems, with the outcome being an extension of knowledge to encompass seemingly "chaotic" phenomena.

Nonlinearity and Holism

Chaos science holds that many systems and behaviors operate in a nonlinear and dynamic manner. May (1989) noted that most biological systems are governed by nonlinear mechanisms; thus, one must expect to see chaos just as often as cycles of steadiness are observed. Within a nonlinear system, radical changes can result from very small and unpredictable sources. Bak and Chen (1991), for example, referred to "self-organized criticality" to describe interactive systems that evolve toward critical states wherein a minor event can lead to a major catastrophe; this concept is analogous to "the straw that broke the camel's back."

Holden (1985) pointed out that most scientists try to control many variables when conducting experiments, while attempting to examine only the effects of changes on a single variable. Thus, they operate under the assumption that sim-plified systems will exhibit relatively simple behavior, which can then be easily analyzed and understood, and finally extrapolated to the whole. Scientists adher-ing to this tradition further believe that, under such circumstances, the results of most experiments should be replicable. Holden maintained, however, that both of these beliefs are often false, and that simple systems can produce very compli-cated behaviors, whereas the same system, under identical conditions, can behave in radically different ways. This phenomenon reflects characteristics of both non-linearity and its related concept of holism.

Holistic theory is necessarily important to understanding dynamic systems (Ber-talanffy, 1968), including chaos science (Bak & Chen, 1991). Holism, of course, is not new to the behavioral and social sciences. The early classical Gestalt psy-chologists objected to microscopic explanations of behavior, and Lewin's related field theory discounted a linear, one-to-one relationship between practice trials and the changes that constituted learning (Hilgard, 1948). Not surprisingly, a reemergence of Gestalt psychology has started to occur within recent years (Rock & Palmer, 1990), with its emphasis on a holistic approach to understanding human behavior and the contributions of subjective experiences in the perceptual processes.

Fogel (1990) asserted that the dynamics of the developmental process are lost when attempts are made to separate the organism from the environment for pur-poses of measurement. To elaborate on this position, Fogel and Thelen (1987) described a systems approach to explain early expressive and communicative activ-ity in infants and young children. The systems approach makes the assumption that order arises in a dynamic fashion as a result of cooperating elements that are changing at different times, rather than resulting from some centrally coordi-nated developmental change that is synchronous across domains. Expressive and communicative action are thus organized as part of cooperative systems with other elements of the infant's development, such as physiology, cognition, behavior, and



20 THE JOURNAL OF SPECIAL EDUCATION VOL 27/NO. 1/1993

social environment. Depending on age, context, and task, the control parameter responsible for development may vary. Fogel and Thelen (1987) also pointed out that, because development is holistic, small changes in the control pattern might produce major consequences for the entire system.

In varying ways, nonlinearity and holism have significant implications for educa-tional procedures and technology employed with persons having disabilities. Holism requires a systems approach to the analysis of delayed development wherein behavior is measured and analyzed from numerous interacting domains. It suggests that behavioral changes reflect multiple variables that not only fluc-tuate across time, but also influence one another when they do interact. This type of "mutual causality" (Schwartz & Ogilvy, 1979) depicts a relationship wherein two or more things or events can jointly affect each other. By moving away from isolated (and often massed trial) training in recent years, the field of developmental disabilities has become more open to holistic approaches in its programs for stu-dents and clients with disabilities. The Individualized Curriculum Sequencing model advocated for students with severe disabilities (Guess et al., 1978; Holvoet, Guess, Mulligan, & Brown, 1980) and the "context relevance" hypothesis (Sailor, Goetz, Anderson, Hunt, 8c Gee, 1988) represent microtheoretical efforts in this direction. Both models emphasize the need to teach skills in "clusters" that encom-pass two or more content domains (e.g., teach communication skills with social and motor skills). Additionally, transdisciplinary team models supported by many persons in the field (e.g., Orelove 8c Sobsey, 1991) have also advocated for a more holistic professional approach to the education and treatment of individuals with significant developmental delays (e.g., Sailor et al., 1989; Sailor 8c Guess, 1983).

On the other hand, the field of developmental disabilities has yet to convinc-ingly address the myriad of variables, including endogenous conditions and fac-tors, that influence organized patterns of behavior and development. This is evidenced in the field's tendency to manipulate only environmental events or con-ditions, and the frequent selection of single independent variables for intervention when attempting to change or develop behavior. Furthermore, there is resistance to nonlinear research models that control for the possibility that even miniscule changes in a single event or condition can radically alter behavioral outcomes, consistent with tenets of chaos science.

Whereas nonlinearity and holism are characteristic attributes of all dynamic systems, chaos theory is accompanied by other important concepts and princi-ples that are derived from extensive research in the physical and biological sciences, theoretical mathematics, and computer graphics.

Attractors

Schoner and Kelso (1988) noted that nonequilibrium systems generally reflect dissipative dynamics, meaning that many independent trajectories within a sys-tem examined under varying conditions will eventually approach each other in space. With sufficient time, these trajectories will converge on a certain limit set that is referred to as an attractor. These limit sets may be points, cycles (oscilla-tions), or neither of these, in which case they demonstrate properties associated




with "chaotic attractors." Limit sets are better understood by presenting the four major types of attractors that have been identified by chaos scientists (and dis-cussed extensively by, among others, Basar, 1990; Crutchfield et al., 1986; and Gleick, 1987).

The fixed or static is the simplest type of attractor and represents a system wherein all trajectories, with time, settle down to a single point. Abraham, Abraham, and Shaw (1990) discussed two types of fixed attractors, nodal point and focal point, which are shown schematically in Figure 1.

The focal point attractor is best described as a pendulum with friction that will oscillate back and forth, but eventually come to rest at a fixed point. Consider, for example, a child with developmental disabilities in a classroom whose gross motor activity level gradually diminishes during the morning hours, comes to a

Focal Point

Nodal Point

Figure 1. Two types of fixed (static) attractors. The focal point attractor shows one trajectory that spirals toward a rest point. The nodal point attractor depicts multiple trajectories toward a rest point with no spiraling. (Adapted from A Visual Introduction to Dynamical Systems Theory for Psychology by F. Abraham, R. Abraham, and C. Shaw, 1990, Santa Cruz, CA: Aerial Press.)




rest point during lunch time, and then shows a repeated oscillation during early afternoon hours. The "friction" responsible for increased reduction in motor activ-ity might result from one or more, possibly interacting, variables, such as fatigue, medication effects, and classroom schedule.

The nodal point (Abraham et al., 1990), displayed also in Figure 1, is an attractor that is approached by multiple trajectories, but without spiraling. An example is the emergence of those developmental milestones that appear with regularity and predictability among normally developing infants. The rest point in the figure could, for example, represent independent walking, a motor skill that is accom-plished by most infants around 13 months of age. Longitudinal studies showing significant age-related declines in measures of intelligence among males with the Fragile X syndrome (Lachiewicz, Gullion, Spiridigliozzi, & Aylsworth, 1987) pro-vide another example of trajectories moving across time toward a fixed point.

Limit cycle or periodic attractors do not, over time, settle down to a single fixed point. Rather they cycle (oscillate) through a sequence of states, yet all trajecto-ries tend to terminate in these cycles. Mood swings (e.g., elation and depression) provide a common example of a two-cycle oscillation. Abraham et al. (1990) dis-cussed these fluctuations in relation to dynamic interactions with measures of posi-tive or negative self-image, as derived from psychoanalytic theory.

Research into behavior state conditions affords one example of studies of cyclic attractors. States have long been studied in normally developing infants (Helm 8c Simeonsson, 1989) and, more recently, among children and youth with pro-found and multiple disabilities (Guess et al., in press; Guess et al., 1991; Guess et al., 1990). State conditions are measures of alertness and responsiveness, rang-ing from deep sleep through crying/agitated states. Wolff (1987), for example, presented a series of extensive longitudinal investigations into the state behavior and emotional development of normally developing infants. His investigations and interpretations are based on assumptions of nonlinearity, consistent with chaos. Similarly, Thelen (1990) discussed behavior states in relation to concepts derived from chaos science, with the various states depicted as attractors.

In their studies of children and youth with profound intellectual and other dis-abilities, Guess et al. (in press; Guess et al., 1991; Guess et al., 1990) used an obser-vation scale that includes the following eight behavior state conditions: deep sleep (S1), shallow sleep (S2), drowse (DR), daze (DA), awake inactive-alert (A1), awake active-alert (A2), awake active/stereotypy (A2/S), and crying-agitation (C/A) that can include self-injurious behavior. In a study underway, 75 children and youth with severe and profound disabilities are being observed for 5 consecutive hours in classroom settings. Event recorders are used to collect continuous behavior state data using the eight categories above, with simultaneous observations taken with a code for environmental and setting conditions. Figure 2 presents state data from one participant (Jerry) who, over the 5-hour observation period, was observed to spend 90% of the time in the awake inactive-alert (A1) state and 9% in the awake active-alert (A2) state. Less than 1 % of his time was observed in the other state categories. The time series analysis in the figure shows a cyclical movement between the A1 and the A2 states. The awake inactive-alert state represents attentiveness and alertness to the environment through the sensory modalities, especially vision



THE JOURNAL OF SPECIAL EDUCATION V O L 27 /NO. 1/1993 23

and hearing. The awake active-alert state includes those behaviors in which the person is actively responding and physically interacting with the environment. The very low percentage of time that Jerry was observed in the awake active-alert state is, unfortunately, typical for children and youth with profound disabilities (Guess et al., 1991; Guess et al., 1990).

During the observation session, Jerry shifted from the A1 to the A2 state 85 times, and from the A2 to the A1 state an almost equal (84) number of times. The top part of Figure 3 shows the average length of time in seconds that he spent in one of the states before moving to the other, and the average length of time that he remained in that state following the shift. Whereas the time series data in Figure 2 show cyclic movement between the two states, the state shift information in Figure 3 demonstrates a definite time duration bias (attractor strength) for the awake inactive-alert (A1) state. This bias is presented visually in the bottom half of the figure, showing cyclic movement between the two attractors. The basin, depicting the region of attractor trajectories (Abraham et al., 1990; Gleick, 1987), is visually presented as being larger for the A1 state to reflect this strength. In this particular example, there are likely several interacting variables (e.g., physical fatigue, task attractiveness) that weaken the awake active-alert state as an attractor, resulting in considerably less time being spent in this state.

Abraham et al. (1990) described the "buckling column" as another type of oscil-lating attractor. Visualize a strong elastic column that extends perpendicularly

JERRY

SEIZ

CA —

A

DA

DR

s2 - I

in iiii II urn i ill II I., I

1 Hr. 2Hrs. 3Hrs. 5Hrs.

Figure 2. A time-series representation of Jerry's behavior state movement across the 5-hour observation period. These data show cyclicity between the A1 (awake inactive-alert) and A2 (awake active-alert) states. Ninety-nine percent of his time was observed in these two states, with less than 1 % of time observed in the A2/S (stereotypy), DA (daze), and DR (drowse) state conditions.



24 THE JOURNAL OF SPECIAL EDUCATION V O L 27 /NO. 1/1993

Jerry

134". 20M

142" 20"

Figure 3. The top part shows the mean number of seconds Jerry was in the A1 (awake inactive-alert) state before moving to the A2 (awake active-alert) and the mean number of seconds that he remained in that state after the shift. The same information is provided for shifts from the A2 to the A1 state. The bottom portion of the figure depicts this information as a two-cycle periodic attractor, with the overall time duration biased toward the A1 state. (Bottom portion of figure adapted from A Visual Introduction to Dynamical Systems Theory for Psychology by F. Abraham, R. Abraham, and C. Shaw, 1990, Santa Cruz, CA: Aerial Press.)




from a base. When this column is pulled to one side and then released, it will swing back and forth for a while, but eventually return to the perpendicular rest-ing point. Suppose that a weight is attached to the top of the elastic column. Pull-ing the column to one side and releasing it will produce again a side-to-side oscillation. The weight, however, will eventually force the column to settle on either the right or the left side. The trajectory will depend on such factors as the initial position of the column, size of the weight, and velocity. Similar analogies can be made to psychological processes, such as mood swings where one mood becomes dominant. Learning to consistently select a correct stimulus in a two-choice, dis-crimination learning task might also produce a biased oscillating pattern (Abraham et al., 1990).

The third major type of attractor, torus, is composed of two independent oscil-lations, also referred to as quasiperiodic motions (Basar, 1990; Crutchfield et al., 1986). This attractor mathematically resembles the shape of a doughnut, wherein the trajectories never completely cover or close the entire torus. Abraham et al. (1990) referred to these types of attractors as force coupled oscillators. This implies that in psychological systems, most oscillators are, in varying degrees, influenced by related biological and environmental oscillators. For example, heightened epi-sodes of aggressive behavior in a child with a severe emotional disability might temporally coincide with monthly (cyclic) visits to an estranged parent (environ-mental event), to a loss of sleep and resulting fatigue experienced during those visits (biological event), or to a combination of the two. In this case, the increases in aggressive episodes are entrained (coupled) to the oscillating visits.

By far the most complicated is the strange or chaotic attractor. It is this type of attractor that is most associated with chaos, and it has in recent years been the subject of considerable scientific inquiry pertaining to its application across varying fields and disciplines. Freeman (1991) used the differences between commuters catching their trains at rush hour and the crowd behavior of mass hysteria as a simplified analogy to help explain chaotic attractors. To a person unfamiliar with train stations, the sight of a large number of commuters might well appear to be random crowd movement, with people running about in every conceivable direc-tion. Beneath this apparent random movement, however, is a complex order wherein each commuter is hurrying to catch a specific train, and where traffic flow can be rapidly altered by announcing a track change. Conversely, mass crowd hysteria is random and remains so; no announcement will make a large mob cooperative. The point is that chaotic behavior appears on the surface to consist of totally random occurring phenomena when, in fact, the randomness is con-strained to varying degrees by an underlying organization or pattern (strange attractor). The problem for scientists is to discover and to validate the pattern or "topographical orbits" of chaotic attractors. The unfoldings of these orbits or trajectories are displayed as phase portraits (Abraham et al., 1990) that visually represent different types of attractors. Trajectories portrayed in the attractors represent the topographical features of dynamic systems and the complex folds and stretches that make up their unique properties. Strange attractors often have names to identify their discoverers, such as Birkhof bagel, Lorenz mask, Rossler band, and Koch curve (Gleick, 1987). Figure 4 shows several of these chaotic attrac-




NAME

Point

Closed Orbit

Birkhof Bagel

Lorenz Mask

I Rossler Band

Rossler I Funnel

PORTRAIT I

• 1

H ^1§§?|

tel \w\

TIME SERIES

hi H

h r\ IV |\A/\ /W\ |

IV v LA AA /̂ l rvrV UU\H i v y v/VJ \ /

Figure 4. Point and closed orbit (oscillating) attractors are compared to several types of chaotic attractors. The phase portraits for the attractors are shown with their corresponding time series graphs for a single trajectory. (Adapted from A Visual Introduction to Dynamical Systems Theory for Psychology by F. Abraham, R. Abraham, and C. Shaw, 1990, Santa Cruz, CA: Aerial Press.)




tors and presents a comparison with point and closed orbit (oscillating) attrac-tors. The phase portrait is shown for each attractor, accompanied by its correspond-ing time series for a single trajectory.

Schmid (1991) described the importance of initial states ("sensitive dependence on initial conditions") to contrast chaotic phenomena with the traditional con-cepts of causality and randomness. Causality implies that, under similar influences, slightly different initial conditions will result in predictable and proportionally small outcomes (e.g., trajectory of a tennis ball). Randomness means that slightly different initial conditions will lead to unpredictable and arbitrarily large out-come differences (e.g., radioactive decay). Chaos implies that similar initial con-ditions will lead to large but somewhat probabilistically determined outcomes, and these outcomes are highly dependent on initial conditions (e.g., behavior of weather patterns). Thus, as mathematical/physical concepts, causality implies statistical certainty and randomness indicates statistical relativity. Chaos, however, represents a statistical probability that, in Schmid's (1991) terms, can be "intuited."

Strange or chaotic attractors thus represent systems or behaviors that have a certain determinism, yet also reflect the influence of random conditions and events that make complete predictability impossible. The unpredictability is due, in large part, to sensitivity to initial conditions wherein "the divergence property of chaotic attractors ensures that a small difference at any given moment in the position of two trajectories will be amplified by expotential growth, and will become a large difference at a later time" (Abraham et al., 1990, p. 11-76). These attractors, neverthe-less, provide a method for identifying some order out of what might initially appear to be total chaos in the behavior of systems.

Importantly, there are also many systems in which chaotic phenomena are likely to be adaptive. Freeman (1991) speculated that neuronal activity in the brain represents a chaotic system that permits humans to display great perceptual sen-sitivity to environmental stimuli, produces novel activity patterns, and may con-stitute the chief factor that distinguishes brain activity from artificial intelligence analogs. Medical researchers similarly conclude that for many healthy physiolog-ical systems in the body, including heartbeats (Pool, 1989b), homeostatic mecha-nisms may best be analyzed as chaotic systems.

Phase Shifts. One of the most interesting aspects of chaos theory pertains to how systems change, moving, for example, from stable and predictable attractors to strange ones. The transition from one attractor to another is referred to as a "phase shift" or, in mathematical terms, a "bifurcation." These bifurcations can create strange attractors out of conventional (i.e., fixed, cyclical) ones, moving from con-trolled causality to apparent turbulence and unpredictability. As stated by Stewart (1989) "bifurcations provide a route from order to chaos and it is by studying such routes that most of our understanding of chaos has been obtained" (p. 47). Phase changes represent qualitative changes in the system where the macroscopic behavior is not linearly predicted from changes at the microscopic level (Gleick, 1987).

Several examples from the physical sciences help explain phase change transi-tions. The boiling point in heating water reflects an abrupt qualitative change




that does not correspond in any linear fashion to increases in water temperature (Gleick, 1987). Similarly, smoke rising suddenly changes from a straight column to a cloudlike turbulence. An example from physiology is represented by horse gait (Fogel & Thelen, 1987). The increased speed of a horse in a linear fashion produces gait patterns (walk, trot, and gallop) that are nonlinear in nature, qualita-tively distinct, and have no stable intermediate stages. Abraham et al. (1990) presented example bifurcation sequences to illustrate how new attractors emerg-ing from these phase shifts result in motivational and personality changes in human behavior.

Fractals

Gleick (1987) related an intriguing story about the mathematician, Mandelbrot, who used a computer to enter, and then analyze, over 60 years of cotton price fluctuations in this country. Traditional economic thinking dictated that cotton prices over time would show small transient price changes that had nothing in common with large, long-term price changes. These short-term fluctuations would, in fact, represent randomness, which needed to be averaged in with the long-term price index to produce the expected normal (bell-shaped curve) with its standard deviation probabilities. Mandelbrot's findings were astonishing. Although each particular price change was random and unpredictable, the sequence of price changes was independent of scale; that is, daily price changes and monthly price changes matched perfectly. The degree of price variation had remained constant over a long time period that included two world wars and a depression.

The cotton prices, in effect, produced a pattern whereby small price fluctua-tions had patterns that were identical to the larger ones. These small identical patterns within larger identical patterns reflect another important aspect of chaotic systems, referred to as self-similarity (Abraham et al., 1990; Gleick, 1987) or scalar invariance (Schmid, 1991).

Simply stated, scalar invariance is the absence of any natural size, time or other intrinsic measure-ment scale. In practical terms, scalar invariance means that a "blow up" of an arbitrary region of the state-space plot (= phase portrait) of the system exhibits the same qualitative structure as is already manifest in the original figure. (Schmid, 1991, p. 191)

Strange attractors are frequently identified mathematically as fractals (Crutch-field et al , 1986; Jurgens, Peitgen, & Saupe, 1990; Schroeder, 1991; Series, 1990), which can be expressed spatially as irregular geometric shapes that possess the same degree of irregularity on all scales (Mandelbrot, 1990). Fractals, therefore, possess the property of self-similarity and, as stated by Gleick (1987), "in the mind's eye, a fractal is a way of seeing infinity" (p. 98). Examples of fractal objects from nature include the fern and the cauliflower where each small branch and frond replicate the geometric pattern of the whole plant at all levels down to microscopic analysis.

Computer-generated fractals from mathematical derivations display magnifi-cent, beautiful, and complex geometric forms that offer their own form of art. More than mathematical calculations, however, fractals delineate the boundary




between regular and chaotic motion (Jurgens et al., 1990; Percival, 1989), and they help to graphically describe the behavior of complex, dynamic systems (Mandel-brot, 1990).

In linear, self-similar fractals, the parts of the object are exactly like the whole. The most important fractals, however, are nonlinear or chaotic; this implies that factors that affect the way the system behaves are not proportional to the effects that they produce. These complex nonlinear phenomena (e.g., weather patterns and changes in insect populations) show behavior typical of deterministic chaos and yet, when displayed as computer graphics, demonstrate the property of self-similarity.

APPLICATION OF CHAOS

Compared with the social sciences, the physical and biological sciences are far advanced in the application of chaos theory. There is, nevertheless, a rapidly grow-ing interest among social scientists in the application of chaotic modeling to the study of human behavior, including issues and challenges related to learning and emotional adjustments. When addressing psychologists on scientific issues pre-sented by relativity theory, quantum mechanics, and chaos, Marr (1990) made the following prediction and challenge:

It (nonlinear dynamics) may ultimately prove to have the broadest influence of all, reaching into every endeavor where significant quantitative methods are applied. Thus, while the mechanical engineer, the physiologist, the economist, the ecologist, the meteorologist, and the behavior analyst need have no practical knowledge of relativity or quantum mechanics, such may not be the case when it comes to nonlinear dynamics. Certainly, those working in fields such as behavior analysis should become familiar, at least in a general way, with the conceptual approaches and issues because they bear upon many aspects—empirical, theoretical, and philosophical—of a behavioral science, (pp. 1-2)

Thus far, applications of chaos in psychology have centered primarily on con-ceptual models, or reinterpretations of traditional theories from the perspective of chaos science. The textbook by Abraham et al. (1990), for example, first presents principles and concepts of chaos derived from the physical and biological sciences. Subsequent sections of the book show how chaos theory can be applied to psy-chology and physiology, includingjungian and Gestalt theory, approach-avoidance conflict, contingent operants, psychoanalysis, cognitive motivational theory, hor-monal networks, and chemical models for investigating schizophrenia.

Schmid (1991) recently presented a conceptual model on the application of chaos theory to understanding schizophrenia, and provided a general hypothesis to directly investigate this relationship. Scalar invariance and corresponding attrac-tors are proposed across three dimensions (sociological, psychological, and bio-logical) to better analyze the onset, assessment, and remission of schizophrenia.

The conceptual approach discussed by Schmid (1991) has considerable relevance to the development and testing of dynamic modeling paradigms for children and youth with disabilities. This would include, especially, the application of chaos theory to new models for assessing and treating the many syndromes and condi-tions (e.g., autism) that display complex, dynamic interactions between organic




and environmental variables and conditions. The application of chaos to inves-tigate more discrete behaviors common to numerous types of disabilities is another logical extension for practitioners and researchers in our field. As discussed earlier, researchers at the University of Kansas (Guess, Mulligan-Ault, & Guy, 1992; Guess, Siegel-Causey, Roberts, 8c Rues, 1992) are currently attempting to apply concepts and procedures from chaos theory to understanding behavior state conditions among children and youth with profound, multiple disabilities. This direction bears similarity to Thelen's (1990) research on normally developing infants, where state conditions are perceived as attractors of varying strength and stability.

Thelen (1990) suggested that the deep sleep state of newborns is a stable or preferred attractor wherein attempts to awaken the infant produce only a tem-porary stirring. Conditions and events that disrupt or change the trajectory of an attractor are referred to in the chaos literature as perturbations. In the case of deep sleep among infants, for example, perturbations might be mild movement, a caregiver's voice, noise, light, and so forth. As a strong attractor, deep sleep is resistant to these perturbations, and the system is likely to return to its original state. On the other hand, Thelen (1990) pointed out that the drowse state among infants is a less stable, or weak, attractor. In this case, perturbations will more easily coax the infant into another state (phase shift), such as deep sleep or wake-fulness. Additionally, in accordance with the predictions of dynamical (chaos) theory, stability is lost when there is a transition between stable states, such that movement from deep sleep to wakefulness is easily detected in the unstable state of drowsiness where the system can be easily perturbed (Thelen, 1990). Wolff (1987) noted that relatively minor perturbations will produce disproportionately large effects in some behavioral states, resulting in either qualitative or chaotic state changes.

The behavior state research by Guess et al. (in press; Guess et al., 1991; Guess et al., 1990) has also included stereotypy as a separate state condition and self-injury as part of the crying-agitation state. Current (Guess, Siegel-Causey, et al., 1992) and previous (Guess et al., in press; Guess et al., 1991; Guess et al., 1990) studies suggest that stereotypic behaviors are potentially strong attractors that are often highly resistant to most perturbations (which often include a variety of envi-ronmentally induced interventions). Although various exogenous events and con-ditions are used to decelerate stereotypy, such behaviors are often resistant to durable change. As part of their analysis, Guess et al. (in press; Guess et al., 1991) used a "lag sequential method" (Bakeman & Gottman, 1986) to identify the prob-abilities that each of the eight state conditions follow one another across time, for example, the probability that stereotypy will be followed by awake inactive-alert behavior, regardless of the overall occurrence of each of these state condi-tions. Results showed that those students with high rates of stereotypy had much tighter sequence patterns than did those students with significantly lower occur-rences of these behaviors. This implies that students with high rates of stereotypy had less predictable movement into the remaining seven state conditions. These students were, in fact, more likely to move back and forth between stereotypy and a small number of other potential states. This finding and current research (Guess, Siegel-Causey, et al., 1992) suggest that stereotypy might, for many students, func-




tion as a very strong attractor that is somewhat resistant to perturbations, and that interacts predictably with only a few other attractors (state conditions), includ-ing self-injury. These findings are important because, consistent with dynamical theory, interventions (perturbations) should be more successful if administered during transitions between strong attractors (e.g., phase shifts) when the system is less stable and, thus, more amenable to change.

Guess and Carr (1991) also noted that self-injurious behavior can possibly be better understood and eventually treated when studied as an attractor, and, likely, a strange attractor. As an example, stereotypy has often been observed to pre-cede the emergence of self-injury among persons with disabilities. Stereotypic rocking (one attractor) might, for instance, suddenly emerge into a qualitatively different attractor, head banging. This phase shift from one attractor to another might be explained as a nonlinear phenomenon in which the emergence of head banging resulted from a small and possibly undetected exogenous or endogenous perturbation at a critical moment in time (e.g., sensitive dependence on initial conditions).

Another potential application of chaos science for the study of both stereotypy and self-injury can be found in the mechanisms of fractals. Fractals, as a display of strange or chaotic attractors, require the availability of extensive longitudinal data. However, fractals reveal important system patterns when they occur, and these patterns have potential diagnostic and remedial implications for self-injury, aggressiveness, or any type of recurring behavior. Fractals supply evidence that what is often perceived as totally random behavior can have an underlying mecha-nism that interacts dynamically with the system under study.

The examples provided here are few in number and likely reflect the relative absence in the behavioral sciences of investigators who structure their studies from the perspective of chaos theory and other nonlinear system models. Nevertheless, the potential for application of chaos to the field of developmental disabilities and special education is unlimited. It includes the full range of learning, social, and behavioral challenges presented by disabling conditions, and the shared as well as unique characteristics associated with various types of disabilities. Our hope is that this paper will serve as an "attractor" for researchers, personnel trainers, and graduate students in our field to explore the application of chaos theory to the assessment, education, and treatment of persons with disabilities.

SUMMARY

Chaos science has rapidly emerged in recent years as a new system theory approach that has attracted the interest of researchers from a large variety of disciplines, including mathematics, physics, chemistry, physiology, psychology, biology, history, and economics. Thus far, chaos has received only limited atten-tion in the behavioral sciences, and virtually no interest from the field of special education and developmental disabilities. The assumptions of chaos that per-tain to nonlinearity, controlled randomness, and dynamic system modeling, offer challenging new directions to the assessment and intervention practices in our field.




Future applications of chaos science to the t rea tment and educat ion of persons with disabilities are l imited only by the creativity of the field. Researchers , the-oreticians, and educators who are willing to familiarize themselves with the rapidly growing body of multidisciplinary literature in this area stand to provide the neces-sary linkages of pedagogical science with emerg ing b reak th roughs in o ther dis-ciplines (e.g., q u a n t u m mechanics in physics). In ou r op in ion , such efforts will eventually redef ine old, and discover new, educat ional technologies appl icable to special educat ion and developmenta l disabilities. If this happens , the r ichness and complexity of h u m a n behavior will be more accurately revealed, and our roles as service providers, researchers , and advocates will become m o r e product ive .

Authors' Notes

1. This paper was supported, in part, by the following two federal grants to the first author: "An inves-tigation of interactions between behavioral state conditions and environmental events among chil-dren and youth with profound and multiple handicaps," (1990), U.S. Department of Education, National Institute on Disability and Rehabilitation Research (Project No. H133G00078), and "Innovations for meeting the special problems of children with severe handicaps in the context of regular education settings," (1990), U.S. Department of Special Education, Office of Special Education Programs (Project No. H086D00013). 2. This paper was supported, in part, by the following two federal grants to the second author: Insti-tute on the Integration of Students with Severe Disabilities (1987), U.S. Department of Education, Office of Special Education Programs (OSEP) (Cooperative Agreement No. G0087C3056), and the San Francisco State University site of the National Research and Training Center on Nonaversive Behavior Management (1987), U.S. Department of Education National Institute on Disability and Rehabilitation Research (NIDRR) (Cooperative Agreement No. G0087C023488).

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