I first became interested in Rene Thom's cusp catastrophe model and it's different applications while researching my book, War in Heaven / Heaven on Earth (Equinox 2005).

The cusp catastrophe is often used to model situations with two independent variables and one non-linear dependent variable [i.e., f(x,y)=z] which exhibit bimodalism, among other things.

Part of the continued appeal of the cusp catastrophe is its qualitative nature; and its wide applicability is due to the fact that it allows you to think about situations in insightful ways.

The basic idea here is the same as that with fear/rage in dogs, and explains why you have to be careful if they are very very fearful, or very very angry, and you are able to undercut that even a little and make them run: The problem is that this can backfire, and they attack you instead. (Note 1.)

I had also come across a dissertation by John Charles Wahl entitled "Faculty Job Attitude: Construction and Analysis of a Series of Models" (University of Florida, 1986) that presented an intriguing application of the cusp model, which I will explain below. The diagrams that I use are from or based on his dissertation.

Generally, John Wahl's job satisfaction model uses the cusp catastrophe to describe the extremes sometimes found among workers' job attitudes, in this case, teachers and faculty. He calls it a "3-dimensional multiple continuum planar interaction model," and it dynamically relates job attitude with (1) the perception of working conditions, and (2) the level of emotional involvement in the job.

Here are descriptions of charts (a) through (d).

(a) Cusp Catastrophe model of Job Satisfaction in Wahl (1986).

(b) Illustrates the transition from a single continuum at low levels of emotional involvement, to a dual continuum at high levels of involvement.

In other words, as emotional involvement increases and you move from the back wall along the surface to the forward edge, job attitude becomes bimodal -- it bifurcates. This is the reason emotional involvement is called the "splitting factor" -- it is the parameter under whose influence the polarization of job attitude occurs.

(c) Viewed from the side, this cross-section shows why reducing emotional involvement decreases the variation in job attitude, eventually converging to a point.

This has a practical application and explains why school administrators are always wary of teachers that are deeply committed to their students (such as new teachers): They are far less stable than those teachers / instructors that have been around a while, with "low" emotional involvement, and they can be set-off by only the slightest changes in their work-setting. This volatility is, by definition, non-linear.

(d) When viewed from above, this shows the narrowing of the cusp (funnel shape) along the "splitting factor," with the solid line on the left representing the outer-most edge of the upper leaf (satisfied job attitude), and the broken line representing the outer-most edge of the lower leaf (DISsatisfied job attitude).

This is as far as Wahl went, but I was able to add a little something to it by applying the "surprise reversal effect" to explain unexpected or SUDDEN job departures -- like when someone comes back to work from vacation for the first time, or (even) has a salary raise, and then quits from being so ticked off (Note 3). The "surprise reversal effect" refers to counter-intuitive outcomes that are even more unexpected than before.

In this case, small changes that REDUCE emotional involvement produce non-linear outcomes due to the narrowing of the cusp in (d), whereas further from the surface edge (solid line), these changes would go unnoticed.