Chaos Theory, Sensitive Dependence, and the Logistic Equation

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 (January 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" (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 (1994), 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 he used a less technical language, Bandura essentially articulated a view of the individual as a sensitively dependent system.