 cusp2
The Catastrophe Machine

## 2. An algebraic version of the double well

It is awkward to calculate precisely the angle at which the catastrophe happens in the double-well example. There is a more algebraic version of the phenomenon which is easier to work with. Imagine that the ball is constrained to roll on the graph of the function y=x4-x2. Instead of tilting the graph we perturb the function by adding a linear term (ax) so as to raise one well and lower the the other. a=0.3 Here is the graph of y=x4-x2+.3x. If the ball had started on the right, it would still be on the right. a=0.6 Here is the graph of y=x4-x2+.6x. The ball would have rolled over to the left. The exact point at which this happens can be reckoned (easy calculus exercise) to be a=(4/3)*(1/6)1/2=.5443...

The corresponding negative values a=-.3, a=-.6 give graphs where the left well grows higher than the right. These values and the initial a=0 generate a family of figures exactly analogous to the the configurations 1 -- 9 of our original double well. The perturbation parameter a plays the role of the angle of tilt.

The catastrophes take place when a=(4/3)*(1/6)1/2 and when x=(1/6)1/2=0.4082... Looking closely at the graphs right near the catastrophe point, for values of a just above and just below the critical value a0=.5443..:

 a=a0+0.00001 a=a0 a=a0-0.00001   makes it plausible that the graph can be reparametrized near the catastrophe point as y=x^3, and that this catastrophe can be explained mathematically as a perturbation of the function y=x^3 so as to create, or destroy, a local minimum near x=0.

 y=x3+x y=x3 y=x3-x   Here we can see the reason for the name "Fold Catastrophe." If the graph is projected onto the y-axis, the catastrophe corresponds to the appearance or disappearance of a pair of folds.

Welcome to the
Feature Column!

These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
Read more . . .

Search Feature Column

Feature Column at a glance