Chaos Theory, Sensitive Dependence, and the Logistic Equation

David R. Mandel

Department of Psychology

University of British Columbia

One commonsense notion about causality is that a slight difference in an antecedent condition should lead to only a minor difference in that condition's effect. In discussing the implications of nonlinear dynamical systems theory, also popularly known as chaos theory, for psychology, Barton (1994) introduced the concept of sensitive dependence, which challenges the preceding notion. According to Barton, "[sensitive dependence] means that if two sets of initial conditions differ by any arbitrarily small amount at the outset, their specific solutions will diverge dramatically from one another over the long range [italics added]" (p. 6). He suggested further that specific behaviors of sensitively dependent systems will be unpredictable over the long range.

Sensitive dependence is sometimes called the butterfly effect--a term that purportedly arose from a 1972 presentation given by Edward Lorenz (a meteorologist and an important figure in the development of chaos theory) entitled, "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?" (Lorenz, 1993). Although at the time Lorenz raised more questions about sensitive dependence in the weather than he answered, the butterfly has become a metaphor for the notion that minor local events can serve to undermine predictability by sometimes significantly altering the course of future events. In psychology, Bandura's (1982) notion that people's life paths can be profoundly influenced by fortuitous events shares many similarities with the concept of sensitive dependence. Just as the butterfly effect depicts how seemingly trivial events (e.g., a butterfly flapping its wings) can have, at critical times, consequential effects on the way a weather system develops, Bandura described how seemingly trivial events in people's lives can have, at critical times, consequential effects on people's life paths. Like Barton, who noted the inherent unpredictability of sensitively dependent systems, Bandura noted that "the unforeseeability and branching power of fortuitous influences make the specific course of lives neither easily predictable nor easily socially engineerable" (p. 749). Although using a less technical language, Bandura essentially articulated a view of the individual as a sensitively dependent system.

To illustrate the effect of slight differences in initial conditions on a process exhibiting sensitive dependence, Barton used the equation,

x' = x + ax(1 - x). (1)

In Equation 1, an initial value of x is used to calculate the next value of x (which Barton denoted as x'); this new value of x is used in turn to generate the next value of x, and so on. The process of feeding the output of an equation back into the same equation as new input is called iteration. Thus, Equation 1 is a feedback process; the string of x values generated by Equation 1 represent the output or behavior of that process for a particular parametric value.

Because of the iterative function of the logistic equation, it is useful to rewrite Equation 1 using subscript notation:

xi+1 = xi + axi(1 - xi), (2)

where the subscript i refers to the step in an iterative sequence. In order to iterate Equation 2 it must first be "seeded" with an initial value, x1, which can range between 0 and 1.

Clarification of Barton's discussion of sensitive dependence and Equation 2 is needed. Following from Barton's definition of sensitive dependence, one might mistakenly conclude that, when a is set at a value such that Equation 2 exhibits sensitive dependence (e.g., a = 3), the numbers generated from two separate iterative sequences of Equation 2 starting out close together would continue to diverge from each other. Although Barton does not actually test this in his article, he suggests contrasting x1,1 = .5000 with x1,2 = .5001 (the second-position subscript refers to the jth iterative sequence or numerical trajectory). I have done so using several different values of a that lead to sensitive dependence. My testing reveals two relevant findings: First, the difference between the two trajectories does not continue to increase (as one would expect if the two trajectories were diverging from each other), but rather it fluctuates and changes sign irregularly. Second, difference values become highly variable earlier in the iteration sequence than one might expect from reading Barton's account (because of his emphasis on unpredictability over the long range). For example, when a = 3, extreme positive and negative difference values can be seen by the 20th step. In short, these trajectories quickly become uncorrelated, despite there being only a minor difference between them at the outset. Barton's claim that, for systems exhibiting sensitive dependence, slight differences in initial conditions will lead to divergent solutions over the long range is not incorrect, because dynamical systems theorists do refer to divergences in this way. His choice of terminology, however, is likely to confuse the unfamiliar reader.

Barton referred to sensitive dependence on initial conditions. First, it is worth noting that initial conditions are sometimes merely arbitrary starting points (as in the preceding example) that do not have any real significance. For example, if children's IQs are measured yearly from ages 5-16, it would be wrong to consider their IQs at age 5 as meaningful initial conditions. These age-5 scores are merely arbitrary starting points for measurement purposes. Second, focusing on initial conditions obscures the fact that the behavior of a sensitively dependent process can be influenced by slight perturbations introduced at any point in time. Thus, it is unnecessarily limiting to focus on initial conditions per se in understanding the effects of perturbations on sensitively dependent processes. Indeed, behavioral scientists may be more interested in noting the fact that a slight change in an input to a feedback process (such as introducing rounding error in the iteration of Equation 2) may lead to an incremental loss of predictive power over time (which is represented by the ith number of steps in Equation 2).

The logistic equation illustrates how a feedback mechanism with a very simple structure can generate predictable or unpredictable behavior depending on the setting of its control parameter. Ward (1993) provided an interesting (but hypothetical) example of how a model similar to Equation 2 could further our understanding of psychophysical scaling by indicating for what parametric ranges psychophysical behaviors should be predictable and for what parametric ranges they should not. Killeen (1989) proposed that the unfolding of human behavior through time shares important similarities with the behavior of the logistic equation, and he used that equation as a model of idiographic-nomothetic distinctions in psychology. The concept of sensitive dependence (and models such as Equation 2) may also shed light on the question of why some individual differences are more stable than others when measured across time (see, e.g., Thelen, 1990). The realization that even the simplest nonlinear processes with relatively few parameters can sometimes account for complex, dynamical behavior that would seem to require many parameters or even multiple systems in order to be accurately described may also prove significant in cognitive psychology. Tulving (1985), for example, proposed a multiple-memory-systems model partly to account for findings of stochastic independence of direct priming effects and recognition memory. He argued that "stochastic independence cannot be explained by assuming that the two comparison tasks [viz., implicit and explicit memory tasks] differ in only one or a few operating components (information, stages, processes, mechanisms)" (p. 395). The present analysis of Equation 2, however, suggests otherwise: Here two identical iterative processes given only slightly different initial inputs generated uncorrelated output behavior. In other words, findings of stochastic independence or lack of correlation between measured behaviors do not imply that underlying mechanisms or processes responsible for those behaviors are themselves nonoverlapping. An alternative possibility is that such processes are nonlinear.

Nonlinear dynamical systems theory may help researchers address what, over 20 years ago, McGuire (1973) called the creative problem in (social) psychology--a general failure to address the complexity of real-world systems and behavior in psychological research. The current excitement about this new science of complexity has led some writers, such as Barton, to describe it as an entirely new paradigm. Although nonlinear dynamical systems theory does offer reasons for interest--even excitement, prophetic claims of the arrival of a new paradigm for psychology may in fact be damaging to the discipline. First, such claims ironically fail to acknowledge the efforts of those researchers, not new to the discipline, who have pursued mathematical psychology all along. Second, visionaries of a new paradigm for psychology may fail to realize that nonlinear, dynamical data-analytic techniques adapted for use in the behavioral sciences are currently scarce and may prove difficult to apply and interpret. Finally, as Ward (1993) noted, it may be more productive to view dynamical and static approaches (as well as linear and nonlinear models) as complementary, rather than as incommensurable paradigms. In short, it is premature at this point to claim that nonlinear dynamical systems theory will offer psychology a new paradigm. Indeed, for most psychologists, who presently lack the mathematical sophistication necessary to fully apply nonlinear dynamical systems theory to problems in psychology, it will probably, at most, offer some new conceptual insights. Whether this fact will change in the future remains to be seen.

References

Bandura, A. (1982). The psychology of chance encounters and life paths. American Psychologist, 3, 747-755.

Barton, S. (1994). Chaos, self-organization, and psychology. American Psychologist, 49, 5-14.

Killeen, P. R. (1989). Behavior as a field of attractors. In J. R. Brink & C. R. Haden (Eds.), The computer and the brain: Perspectives on human and artificial intelligence (pp. 53-82). North-Holland: Elsevier Science Publishers.

Lorenz, E. N. (1993). The essence of chaos. Seattle, WA: University of Washington Press.

McGuire, W. J. (1973). The yin and yang of progress in social psychology: Seven koan. Journal of Personality and Social Psychology, 26, 446-456.

Thelen, E. (1990). Dynamical systems and the generation of individual differences. In J. Columbo & J. Fagen (Eds.), Individual differences in infancy: Reliability, stability, and prediction (pp. 19-43). Hillsdale, NJ: Lawrence Erlbaum Associates.

Tulving, E. (1985). How many memory systems are there? American Psychologist, 40, 385-398.

Ward, L. M. (1993). Law and order in psychophysics: The role of nonlinear dynamical models. In A. Garriga-Trillo, P. R. Minón, C. Garcia-Gallego, P. Lubin, J. M. Merino, & A. Villarino (Eds.), Fechner Day 93 (pp. 254-259). Madrid, Spain: Universida Nacional de Educación a Distancia.

Author Notes

I thank Lawrence Ward for his helpful discussions of nonlinear dynamical systems theory. I also thank Darrin Lehman, Carol Martin, Larry Vandervert, and Melissa Warren for their helpful comments on an earlier draft of this commentary.

Address correspondence to David R. Mandel, Department of Psychology, University of British Columbia, 2136 West Mall, Vancouver, B.C., V6T 1Z4. Email: dmandel@unixg.ubc.ca.