Catastrophes
The animation shows the function f(x) = x^{4}-x^{2}-c x, where c varies between -1 and 1. The movie starts when c=0 then goes to 1, returns back to 0 goes to -1 and back to 0. The system always choses the nearest local minimum as mandated by Delay assumption of catastrophe theory. Parameters (=time values here), where a local minimum disappears, is called a catastrophe. The picture to the right is the "bifurcation diagram". The vertical axes is the c axes. The movie starts in the middle, then c goes up and down. For every c, the minima (blue) and maxima (red) are drawn. The background song is "Smooth operator", recorded in 1984 by the English group Sade. Even a "smooth operation" can produce catastrophes. More ... |
An application in psychology: multistable perception. Start with the picture to the left and move to the right (the c parameter is here is the -1 at the left of the picture and 1 at the right of the picture. Your perception will change. You see that as in the movie, catastrophe happen at a different point. This process is modeled very well by the above function. This example from multistable perception is discussed for example in the book "Catastrophe theory and its application" by Tim Poston and Ian Stewart. (The picture sequence was redrawn in vector graphics).