SYSTEM BEHAVIOR
AND THE CURRENT EDUCATIONAL PARADIGMS

CHAPTER TWO. SYSTEM BEHAVIOR AND THE CURRENT EDUCATIONAL PARADIGMS

• The Issue of Interaction - Four Views
  • Philosophical Pragmatism
  • Research Reproducibility
  • Research Quandary
  • Turbulent Practice
• A Taxonomy of System Interaction Behavior
  • Definitions of Determinism
    • D-1. Simple Interaction Models
    • D-2. Complex Interaction Models
    • D-3. Indeterministic Interaction Models
  • D-1. Simple Model Subdivisions
    • The Convergent/Divergent or Price's Paradox
    • D-1. Systems
      • Fixed point.
      • Limit cycle.
      • Torus.

  Chapter Two - Part 2

• Convergent Taxonomy and Social and Educational Inquiry
  • Fixed Point Perception
    • Social and Statistical Science
    • Educational Science and Counter Movements
      • The mainstream.
      • Counter-movement and negation.
  • Limit Cycle and Torus Perception
• Overview

CHAPTER TWO. SYSTEM BEHAVIOR AND THE CURRENT EDUCATIONAL PARADIGMS

The Issue of Interaction. Four Views

Philosophical Pragmatism

Phillips criticizes the case of Cronbach (1957, 1975, 1977, 1982, 1986) and Snow (1977; Cronbach and Snow, 1977) for using concerns with interaction to undercut the predictability of the scientific method. Cronbach and Snow make three general arguments based on the long study of aptitude-treatment interactions (ATI):

1. There is significant interaction with ATI and other related variables; time is also a significant variable in interactive effects; so for both instances what is true now may not be true in so many years.

2. The consideration of even a few interacting variables in a simple area of interest produces complexities beyond our ability to calculate or even collect the relevant data. In short, interactions in general are detrimental to research effectiveness and interactions of a few variables produce incalculable complexity. Gage critically summarized the implications as: "Thus we are told, the best we can hope for is not prediction and control but mere temporary understanding" (Gage, 1978, p.91).

3. Further, at best, theories will only apply to very specific contexts.

In summary, the problems created by interaction are decay, complexity and specificity.

Phillips (1987) counters with three arguments of his own. His first point (1a.) is that interaction is not new and has been overcome before, and so may again in the context of social science. That is, the fact that entities interact does not in and of itself prevent scientific progress, useful progress as defined by Gage's penchant for prediction and control. For example, the pressure, volume and temperature of gases interact, but Boyle's law has proved an effective predictor. Also, (1b.) the interacting factor of time with potential decay of generalizations has been countered effectively in the biological sciences. Finally, (1c.) there are aspects of the natural world that are too complex for accurate prediction (predicting when a leaf in a storm will hit the ground) but theories are attainable and useful. Second, Cronbach and Snow's contention that facts decay before they can be used to deduce theory is poor science. Popperian science (good science) begins with a good hypothesis which significantly reduces the number of variables and the number of interactions to which the researcher must attend, bringing the problem down to calculable levels. Third, contrary to narrow specificity, theory at some more general level will always be possible and may require prior, more local theory, from which to work (e.g. Darwin's theory about change and mutation).

In analyzing Phillip's argument, it is noteworthy that he does not counter and thereby appears to accede two important points of Cronbach and Snow: 1. interactions cause difficulties for prediction and control in research; 2. generalizations can decay. Further, he makes no attempt in this criticism to define interaction nor distinguish any variation to the concept.

Shifting now from exposition to criticism, a number of problems appear with Phillip's rebuttal. His argument in his first point uses forms of a science to which both he in this book and his philosophical mentor Popper have gone to considerable length to criticize, logical positivism and inference. The weakness of the inference of his arguments 1a. and 1b. is that they await only one counter-instance for falsification. Chapter three will offer just such falsification, instances where the interacting factors of time and decay appear to be insurmountable. His second point regarding the focusing effect of hypotheses is only valid if the problem is truly complexity caused by the number of variables. If the simple calculable examples in chapter three and an example presented later in this chapter stand the test of time, Phillip loses this point also. Finally, there is little reason to contend with his third point (similar to 1c.), shifting to more general levels, for it can cut either way. One's theory might be that there is no theory or that prediction and control is useless. The point contributes little to his claim without some claim for what he wants theory to do.

Phillip later criticizes another line of argument related to interaction, Hegelian philosophy. Hegel and others make a distinction in types of interaction, external or internal. External interactions incorporate factors whose internal nature is not affected by interacting. Some of the factors involved in ATI research are of this type. Age and sex (and arguably IQ and SES) do not change with the treatment. "In contrast, however in internal relationships the interacting factors are themselves partly constituted by the interactions, so that these factors are different when interacting from what they are when independent or isolated" (Phillips, 1987, p.67). The example given for internal interaction is the change in the weight of a stone from the earth to the moon. I find the example weak in comparison to say the relationship between the populations of hawks and mice in a given area. However, Phillips does note that many believe "the social domain is riddled with internal relationships" (Phillips, 1987, p.68).

The holism of general system (GST), says Phillips (1987), depends on the concept of internal relationships for dynamically interrelated systems, stating that the removal of ANY one variable or the disconnection of any interaction would make the system different from what it was, precluding reductive science from producing meaningful results. This is part of the position taken by Lincoln and Guba (1985) and Dewey with his reflexive arc. William James ridiculed this idea that "everything is self-created through its opposite - you go around like a squirrel in a cage" (James, 1884, p.283).

Bandura invented his own terminology for internal interactions, the reciprocal determinism of his cognitive learning theory (Bandura, 1977, 1978). Bandura also places great emphasis on the complexity of such multiple relationships. Phillips counters that in principle, events still occur in separation over time so that without the belief in simultaneous events, interactions can be shown to "be a series of unidirectional causal influences" which by their nature are amenable to experimental control (Phillips, 1987, p.74).

Phillips' primary objection to holism is that the part scientists wish to investigate will have some characteristics that are defining and some that are secondary. Therefore, part A can be isolated without changing the nature of A because the relationships with the system may (my emphasis) be secondary.

The case for holism states that if any one strand (interaction) is eliminated the whole becomes something else. Phillips argues that if there are any secondary or lessor links, the argument for holism is lost. A less rigid case for holism, however, is possible and defeats Phillips's simplistic case. In a less rigid system, only some of the key factors may be used some of the time or one allows for some or most of the links to be broken yet still have a system greater than the sum of its remaining parts. Further, the isolation of part A may reveal useful predictable aspects about part A but still not reveal what will happen when part A is connected to a larger system.

Phillips does not raise the level of his argument by passing along the view that many believe that "Cronbach and Snow are becoming more addled with each passing day" (1987, p.84) nor by passing on William James's taxonomy of tough minded empiricists versus sentimental soft-headed tender-hearted idealistic reconceptualists. But, it should be noted here that the hypotheses under consideration (the compellingness of chaotic dynamics for educational interaction and the fit with educational concepts) will receive a base of "tough-minded" experimental evidence from the "hard" sciences, not simply argument based on "soft-in-the-head" philosophy.

Overall, Phillips finds the issue of interaction central to the discussion of paradigms, yet an overdramaticized problem that only requires a shift from the logical positivist paradigm to Popperian post-positivism. Later, we shall deal with those who find causal models inappropriate for human affairs. For now, the important concept to retain is that Phillips's rebuttal contains serious difficulties. The narrative will carry forward Hegel's valuable concepts of internal and external interaction and turn to another who has viewed the problem of cause and effect.

Research Reproducibility

Though citing Cronbach often, including specific acceptance (p.119) of Cronbach's concern that generalizations decay, Krathwohl has no response to the fundamental problems related to this concern that were tackled by Phillips. He proceeds with conviction to articulate issues that will promote Campbell and Stanley's experimental and quasi-experimental design, believing that "Cronbach's simplification of the whole field to the powerful concept of reproducibility is elegant" (p.278) and states that "gradually the uncertainty about a finding is reduced as these studies accumulate, and a broader and deeper consensus develops among the members of a discipline about the nature of a phenomena" (22). Krathwohl does not resolve the contradiction between his acceptance of both the decay and accumulation of generalizations. For both ideas to be true or believed would require a very equilibrium based perception of educational system behavior, a perception whose grounds for belief will be examined later.

Krathwohl develops a science whose goal is to confirm, with the chain as a guiding metaphor, each link of the chain a confirming step or experiment. This contradicts previously discussed Popperian science which claims that inference is incapable of confirmation and that science can only falsify. Though Popper is acknowledged and accepted, human psychology appears to require, in his view, a perspective that seeks confirmation.

His "cloudlike and clocklike" metaphor for interaction complements the previously described ideas of internal and external relationships. He further contributes the concept of probability to counter the loss of exactness or precision of clocklike worlds. More importantly, he adds to our perception of interaction with a simple taxonomy (p.215-216) consisting of unidirectional (both simple (A.) and multi-casual (B.)) models and a feedback model (C.).

A. simple	B. multi-causal	C. feedback
a--> b	a1-->b1	a1 --> b1
	c1-->b1	b1 --> a1
Figure 2-1. Krathwohl's Models of Interaction

The concept of optimization also appears in Krathwohl's argument, (Figure 2-1) noting that feedback from two or more criteria or variables can seek the best midpoint across multiple goals, a clear indication of a perspective seeking equilibrium. The concept of feedback builds a more dynamic view of educational systems: "For example, structuring the learning environment and tasks increases achievement but suppresses creativity. The teacher seeks to provide enough structure to increase achievement as much as possible consistent with stimulating as much creativity as possible" (p.225). Yet the perception here is a settling down of variation until a stable happy medium is found, an eventual end to the dynamic.

Krathwohl's concepts of probability, optimization and model taxonomy will be carried onward into later sections. The resolution of the puzzling psychological contradiction between the acceptance of decay and accumulation in generalizations will also await later work. Krathwohl's view of interaction represents those who have not found in the nature of interaction any saber-rattling rationale for changing the research status quo.

Research Quandary

The primary cause for its failure are unresolvable problems with aspects of observation: 1. complex interpretive judgment within observers; 2 agreement between sets of observation is limited; 3. ties between observation and concept is limited. He also attributes the larger problem to one of complexity through the high number of variables involved.

Vialato however does not contend with Phillips' counter that pragmatically there is nothing in principle to prevent progress in observation, nothing that prevents the observers' work from being re-tested and re-hypothesized to allow for better progress through simplification. But Violato's goal is to predict personality and this is not possible without consistency. Unfortunately, the fact that "the bulk of data to date indicate that indi-viduals do not demonstrate either transtemporal or transsituational consistency in behavior" (Violato, 1988, p.17). Violato cites a number of supporting studies: Epstein, 1980; Lamiell, 1981; Mischel, 1968; Mischel & Peake, 1982; Rorer & Widiger, 1983; and Travis, Violato and White, 1982. He calls for further work to break through this problem.

Figure 2-2. Violato's Four Models of Person-Situation Interaction

Also of interest to this narrative is Violato's taxonomy that claims to cover all the variations in interaction research. (See figure 2-2. Violato's four models.) It includes both the unidirectional and feedback loop models of Krathwohl but introduces new variations related to feedback. In this taxonomy (p.8), B equals behavior, S equals situation and P equals personality. These models represent an important expansion in the taxonomy of this narrative, in particular not only an expanded view of multi-causal models, but models which incorporate feedback as integral to the design (models 3. and 4. of Figure 2-2).

Endler's work (1966, 1973) would represent the kind conceptualized by model 1. unidirectional interaction. Bowers (1973) envisions the person and environment being interdependent, model 2. The third kind of interaction, reciprocal or dynamic interaction, is presented in two forms, 3. and 4. with some of Mischel's work after exploring ATI research as "(t)he more complex models of interactionism representative of this" (1977, 1979).

The way Violato chose to further analyze these categories is revealing. Vialato misses the significance of a conclusion he states which explicitly ". . . posits mutual or reciprocal influence or determination between persons, situations and behavior, incorporates a remarkably abstract sense of causality and seems to lose track of what specifically is caused, influenced or determined, and what is constant through time" (p.9). As a consequence he confines his further analysis to only the figure 1. category of Endlerian (Endler, 1973) unidirectional analysis of variance-based conception of interactionism. Had he continued the wider analysis, he might have considered a potential relationship between the reciprocal influence models (Figure 2-2) and his data which refuses to act consistently.

In summary, Violato's view represents one of frustration and disappointment in work that investigated interaction.

Turbulent Practice

Schon's reaction is to move from a command or hierarchical posture to one of participant in a heterarchy and to call for reciprocal reflection-in-action. Again, the chaotic nature of the interaction is attributed to the complexity of the large number of variables of the current social scene.

For Schon, change and mutation are not phenomena to be reduced or eliminated, but ones to be accommodated. This is a proper response to insufficient knowledge for clear decision coming from philosophy and research. Whether it must be the only or the primary response depends on further discussion of the potential for scientific prediction and control in the study of system behavior.

In summary of this general review of interaction, the literature has produced several widely different conflicting views on the role and importance of interaction. Two conclusions are drawn: 1. in spite of these diverse views, there is consensus that interaction is a fundamental point of reference; 2. this debate lacks a rigorous framework for making distinctions about interaction. The next section will explore to what degree further distinctions can be made with dynamic interaction models.

A Taxonomy of System Interaction Behavior

Definitions of Determinism

D-1. Simple Interaction Models

D-2. Complex Interaction Models

LINEAR				NONLINEAR
Equation	One Equation	Several Equations	Many Equations	One Equation	Several Equations	Many Equations
=======	=======	=======	=======	==========	=======	=======
Algebraic	Trivial	Easy	Essentially Impossible	Very Difficult	Very Difficult	Impossible
Ordinary	Easy	Difficult	Essentially Impossible	Very Difficult	Impossible	Impossible
Partial Differential	Difficult	Essentially Impossible	Impossible	Impossible	Impossible	Impossible

Figure 2-3. Ease of Solution (Hall & Day)

A more realistic example of complexity as it is commonly perceived would be a model of an ecosystem which "normally includes from a dozen to several hundred or even more simultaneous equations, and they are as likely to be nonlinear as linear" (Hall & Day, 1977, p.9). It would also be possible to conceive of the classroom as an ecosystem of similar complexity. Phillips, Violato and Schon saw complexity from this perspective.

However, the computer has also helped to reveal levels of complexity not adequately articulated by Hall and Day's table (figure 2-3). The top left corner of this table implies that there is a class of problems of one and several equations that are respectively trivial or easy. Alas within this apparently tractable class there exists robust other classes of problems that defy solution. Yet, for such problems, the inputs are finite and precisely known, algorithms are known and effectively executable, the number of variables few, and as the table states, linear.

This new complexity taxonomy has emerged from a field within computer science that began in the 1960's and is devoted to the study of algorithmic complexity. Their taxonomy reveals three classes of problems not computable in practice. The lowest level contains intractable problems that are only computable in principle, but whose solution time or storage space involves quantities far larger than the known life time of the universe or greater than the number of estimated protons in existence. As of now this class consists of nearly 1000 diverse algorithmic problems (Harel, 1987). Two other classes exist above this mere intractable level, the undecideable and the highly undecideable. A conclusion that can be drawn from this analysis is that there will not be a technological fix for a broad class of problems, a class for which there is absolutely no uncertainty about measurement. For this broad class, it is simply irrelevant that the number of components on a chip has doubled every year since at least 1965 and is likely to be 100 million by the year 2000, that mass storage doubles while the cost drops by half every two years and that the speed of computers has increased by a factor of ten every year for the last ten years. This account helps to set the stage for a later class of deterministic problems that must contend with measurement uncertainty. Further, as the reconceptualists have apparently not managed to work this class of problems into their argument for indeterminism, it would seem likely that this class of complexity has not been noticed by the general educational literature.

D-3. Indeterministic Interaction Models

The point of noting these distinct views towards determinism is to enable the reader to understand which line of argument is being taken for the sake of further rejection or acceptance. This thesis bases its case primarily on D1 models. As shall be shown next, simple models (D1) do not always produce simple practical real world solutions.

D-1 Simple Model Subdivisions

Earlier, Krathwohl gave us the example of a teacher trying to optimize between creativity and structure. The logistic equation, though first modeled on populations of animals such as the lynx and the hare, will be used with our teacher's setting in mind. The equation is a feedback loop even simpler than Krathwohl's example (figure 2-1, B.) with but one variable in feedback with itself, but with the added complication of two constants or it could be seen as the simplest form of mutual interaction in Violato's scheme, (figure 2-2, B.). In BASIC computer language, the equation introduced very briefly in chapter one looks like this:

5 LET X = 0.5: LET E = 2

10 LET X = X * E * (1-X)

20 GOTO 10

In this model, the initial value of X represents some initial level of creative activity, say 0.5 on a scale from 0 to 1 with 0 representing activity with no creativity, totally dominated by structure. The sub-operation (1-X) represents a counter-balance on the system. The more creativity, the more disorder there is in the classroom, defeating aspects of creativity and requiring the institution of structure. The variable E (scale from >0 to 4) in this instance represents the amount of energy brought to the effort of dealing with these creative teaching activities. Further note that this model represents internal interaction, but a very simple form where X is a function of itself and X bears external relationship or interaction to two variables (constants), that is, changes in X do not change E or the value 1. The asterisk (*) is the multiplication sign. Each iteration of the loop from line 20 to line 10 represents some time scale, perhaps, one day.

The variations possible with this simple model are enormous though potentially misleading. With the energy level of E = 2, the teacher in this model will raise creativity in the class to what becomes a stable steady equilibrium level. Without further test of this equation, the steady state could be misleading. At higher levels of E the teacher would experience one day with high levels of creativity and low the next (or high structure), a two level periodic solution. From these simulations, one might conclude that prediction is possible. But at still higher values of E, say 3.9, the levels of creativity in the class vary from one day to the next in what appears to be a random sequence of never repeating chaotic numbers, that is without periodicity or equilibrium representing all variations between extreme structure and extreme creativity. A further potentially misleading aspect comes from a further complication; very particular high values of E will produce orderly three level periods and there are other periods hidden within the chaotic output range of particular high values of E as well. Therefore, complexity can come from the nature of the interaction, not just from the number of variables. (Also worth consideration is the question of how one can claim that this equation represents any reality, let alone that of the classroom, which is an issue that must be addressed later.) The primary point of this exercise is that there are surprising aspects of very basic interaction models that have simply not been commonly explored and need such exploration (May, 1976; Ford, 1989).

The Convergent/Divergent or Price's Paradox

This logistic model appears to raise a paradox of fundamental proportions, that, after its creator, I call Price's paradox. Its resolution is essential to further conceptual progress. On the one hand my statements imply that this deterministic model in its convergent phase of periodicity and equilibrium can be predicted while in its chaotic phase it cannot be, that this deterministic model is indeterministic. Can we on one hand believe with Einstein that "Nature does not throw dice", that nature is self consistent and thereby believe in determinism and on the other hand also deny that simplistic interpretation is possible (hence not believe in determinism)? How can this be? It is logically absurd to simultaneously believe and not believe. "The idea that a system can be both deterministic yet unpredictable is still rather a novelty. (We are dealing with something quite different from quantum uncertainty here, which is based on indeterminism)" (Davies, 1989, p.6). Let me assure those of you who may race to your personal computers to observe this miracle, that at unchanging initial values, your computer will (or at least should, if it is working correctly) deterministically produce precisely matching values each time the computer program is run. The problem, the indeterminism, arises not with the equation or with the computer but with measurement and the nature of nature. Very small changes to decimal places of higher values of E can shift the system from chaotic to periodic behavior or back to chaos. (The next chapter will consider further educational precision and the general sensitivity to precision in chaotic models.) When these subtle value changes lie beyond our measurement precision, the deterministic system produces indeterministic results for human beings, but not indeterministic results for nature. Nature "knows" and uses the precise values of an infinite number of decimal places and thereby determines what humans cannot determine. From nature's perspective there is no paradox; the paradox is ours alone.

Figure 2-4. State Space

State space is a useful concept for visualizing the behavior of dynamical system. It is an abstract space whose coordinates are the degrees of freedom of the system's motion. The motion of a pendulum (top), for example, is completely determined by its initial position and velocity. Its state is thus a point in a plane whose coordinates are position and velocity (bottom). As the pendulum swings back and forth it follows an "orbit," or path, through the state space. For an ideal, frictionless pendulum the orbit is a closed curve (bottom left); otherwise, with friction, the orbit spirals to a point (bottom right). Note. From "Chaos" by James P. Crutchfield, J. Doyne Farmer, Norman H. Packard and Robert S. Shaw, 1986, Scientific American, 255 (6), p.49. Copyright 1986 by Scientific American. Adapted by permission.

The next chapter will explore the implications of other such models as the logistic equation for other disciplines and the potential of this perspective for education. The goal now for this chapter is to turn to the study of basic system behavior and convergent systems and their historic role in directing educational research and perception.

D-1. Systems

Two very basic concepts will compose the operation of a dynamic system:

...(T)he notions of a state (the essential information about a system) and a dynamic (a rule that describes how the state evolves with time). In general the coordinates of the state space vary with the context; for a mechanical system they might be position and velocity, but for an ecological model they might be the populations of different species. (Crutchfield, Farmer, Packard, Shaw, 1986, p.49)

Geometric forms, points in a plane, have been developed that characterize long-term behavior of such systems. The general tendency of the motion of these points towards certain patterns, earned them the title of attractors. (See figure 2-4: state space.) "Roughly speaking, an attractor is what the behavior of a system settles down to, or is attracted to" (Crutchfield et al, 1986, p.50). This may take fractions of a second in some phenomena to eons in other systems.

CONVERGENT			DIVERGENT
FIXED	CYCLE	TORUS	CHAOTIC (STRANGE)
Figure 2-5. Taxonomy of Interaction Chart

Within the framework of the theory of dynamic systems, a taxonomy of the motion of interaction has emerged. These categories will be divided into two sets (see figure 2-5. Taxonomy of Interaction Chart), convergent and divergent. For the convergent division of the taxonomy, the categories are: fixed point, limit cycle (closed orbit), torus (bagel) (See figure 2-6: convergent attractor states, next page.) For now, the categories of the divergent division will simply be called chaotic or strange. These motions can be visualized by phase portraits, time series or spectrum analysis (See figure 2-7: attractor table, next two pages). Four texts by Abraham and Shaw provide clear introduction to both divisions of these concepts (1983, 1984, 1985, 1988).

Figure 2-6. Convergent Attractor States

Attractors are geometric forms that characterize long-term behavior in the state space. Roughly speaking, an attractor is what the behavior of a system settles down to, or is attracted to. Trajectories from the initial states eventually approach the attractors. The simplest kind of attractor is a fixed point (top left). Such an attractor corresponds to a pendulum subject to friction the pendulum always comes to the same rest position, regardless of how it is started swinging. The next most complicated attractor is a limit cycle (top middle), which forms a closed loop in the state space. A limit cycle describes stable oscillations, such as the motion of pendulum clock and the beating of a heart. Compound oscillations, or quasi-periodic behavior, correspond to a torus attractor (top right). All three attractors are predictable: their behavior can be forecast as accurately as desired. Note. From "Chaos" by James P. Crutchfield, J. Doyne Farmer, Norman H. Packard and Robert S. Shaw, 1986, Scientific American, 255 (6), p.50. Copyright 1986 by Scientific American. Adapted by permission.

Subdivisions and further discussion of the divergent strand will be held over for the next chapter. All the categories of the convergent division (fixed, cycle and torus) are understood to be amenable to scientific prediction and experimental control. All models from both divisions can be constructed simply, that is with a minimum of variables.

Fixed point.

Limit cycle.

The metronome and the clock are two practical mechanical applications. Electronic oscillators are another example. Note that in limit cycle systems, the energy levels driving the system are higher and sustained as opposed to the absence of continued multiple forces in fixed point systems.

There are also important chemical and biological examples that occur of a spontaneous or self-organizing nature. One of the first to be observed chemical cycles occurred with the oxidation of citric acid by bromate in the presence of cerium ions in 1958 (Belousov, 1958). Another important discovery was the set of periods in the glycolytic pathway (Hess & Boiteux, 1971). In the brain, there is the acquisition, transfer and processing of information through neural oscillators (Meech, 1979). Cycles by day, month or year have been observed in biological systems. The heart and the predator and prey relationship have also been seen as biological examples of the limit cycle. "A high population of prey one year, for example, will lead to an increased population of predators the next year. ...The high predator population results in a reduced prey population the next year, and so the populations tend to cycle up and down through time" (Ford, 1987, p.22).

There is ample evidence too for human circadian rhythms (Wever, 1979) as well as a wide range of other time divisions. For humans and other biological systems there appear at least five functional advantages of periodic regulation: temporal organization, spatial organization, prediction, efficiency and precision of control (Rapp, 1987). This functional advantage will be further considered later when exploring the potential for periodic behavior in educational phenomena.

Torus.

Because all of these three basic forms of system behavior (fixed, cycle and torus) lend themselves to prediction and control, the term convergent, will be generally used to refer to them. Their convergent nature can produce reliably deterministic behavior for both short and long term prediction.

Figure 2-7. Attractor Table

Note. From The Visual Mathematics Library. Dynamics-- The Geometry of Behavior: Part two: Chaotic Behavior (p.131) by R. Abraham and C. D. Shaw, 1983, Santa Cruz, CA: Aerial Press, Inc.. Copyright 1983 by Aerial Press. Reprinted by permission.

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