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Priel and Schreiber's concise article invites one to consider several complex psychological variables in relation to non-linear dynamics (catastrophe theory, chaos theory), a branch of mathematics that covers phenomena that appear random but are actually deterministic. When the equations which describe the behavior can be discovered, they can be modeled. This kind of modeling is made possible through the advances of modern computers. Consequently, much of the terminology and imagery presented in the article deals in part with geometric diagrams, a subject not commonly seen in psychological articles.
The authors begin with “bifurcation points” which they define as those points which “designate the emergence of several new and stable solutions.” They introduce a Feigenbaum diagram which demonstrates bifurcation points; the diagram expresses how the frequency of some oscillatory phenomenon is rapidly doubling, and the system is shifting into chaos. The authors highlight this critical time period (or so-called phase state shift) because this “moment” involving the evolution of chaos is something about which experts in chaos theory generally agree: from a strictly mathematical perspective “universal” principles and mathematical constants characterize the changes, apparently independently of the nature of the system involved (whether laser, heart, electronic circuit, weather, or whatever). Although the subject of chaos (and its graphical representation) can indeed be esthetically beautiful, the subject nevertheless remains dauntingly difficult, and finding mathematical solutions to chaos-related functions is at present sometimes impossible. Also, there will be questions about whether and how to apply this specialty knowledge to psychoanalysis.
Within chaotic systems in general initial conditions can prove decisive, making a difference, for example, in which direction is taken at each point of bifurcation. This is called the “butterfly effect” according to the example in whi